Several complex variables is the natural extension of single-variable complex analysis to higher dimensions, but the theory diverges dramatically from what one might expect. Unlike holomorphic functions of one variable, functions of several complex variables exhibit rigid geometric constraints that profoundly shape the subject. This course explores the foundational theory of multivariable holomorphic functions, beginning with their basic properties and building toward the central insight that domains in higher dimensions possess intrinsic geometric obstructions to [analytic continuation](/page/Analytic%20Continuation)—obstructions absent in the single-variable setting. The course is organized around two intertwined themes: understanding the structure of domains where holomorphic functions can exist, and developing the machinery needed to solve fundamental existence problems. The first chapters establish holomorphic functions of several variables and introduce the Hartogs extension phenomenon, which reveals that [analytic continuation](/page/Analytic%20Continuation) can occur in unexpected ways. From there, the focus sharpens to domains of holomorphy—the natural domains in which holomorphic functions live—and introduces plurisubharmonic functions and pseudoconvexity as the geometric tools needed to characterize these domains. These concepts are the multivariable analogs of subharmonicity and convexity, but with crucial differences that reflect the richer geometry of higher dimensions. The later chapters develop the analytical machinery to solve classical existence problems. The $\bar\partial$-equation and Dolbeault cohomology provide a bridge between geometry and analysis, allowing one to construct holomorphic functions with prescribed properties. The Cousin problems ask when certain local analytic data can be assembled into global holomorphic functions, and their solution—via the Levi problem—shows that pseudoconvexity is both necessary and sufficient for such gluing. The course concludes with Reinhardt domains, whose logarithmic coordinates simplify the geometry, and an outlook on biholomorphic inequivalence, illustrating how dramatically different domains of holomorphy can be despite sharing topological features. # 1. Holomorphic Functions of Several Variables This chapter launches the course by laying the foundational theory of holomorphic functions in several complex variables. We begin with the basic definitions — Cauchy–Riemann equations, [power series](/page/Power%20Series), the polydisc — and discover immediately that several variables behave very differently from one: separate holomorphicity unexpectedly forces joint holomorphicity, isolated zeroes are impossible, and the geometry of zero sets is fundamentally richer. The central results are [Osgood's lemma](/theorems/3379), which establishes the equivalence of local boundedness and joint holomorphicity for separately holomorphic functions, and [Hartogs's theorem on separate holomorphicity](/theorems/3400), which shows that separate holomorphicity on a product domain is already sufficient for joint holomorphicity. ## Holomorphic Functions on Open Sets in $\mathbb{C}^n$ The space $\mathbb{C}^n$ consists of $n$-tuples $z = (z_1, \dots, z_n)$ with each $z_j \in \mathbb{C}$. Writing $z_j = x_j + i y_j$, we identify $\mathbb{C}^n \cong \mathbb{R}^{2n}$ as a real [vector space](/page/Vector%20Space). An [open set](/page/Open%20Set) $\Omega \subset \mathbb{C}^n$ is an open subset in the Euclidean topology of $\mathbb{R}^{2n}$. The question of what it means for a function $f: \Omega \to \mathbb{C}$ to be holomorphic admits three a priori different answers — Cauchy–Riemann equations, complex differentiability, and local [power series](/page/Power%20Series) expansion — and [Osgood's lemma](/theorems/3379) establishes their equivalence under a mild local boundedness condition. We develop each in turn. ### Cauchy–Riemann Equations and Complex Differentiability What should it mean for a function $f: \Omega \to \mathbb{C}$ to be holomorphic when $\Omega \subset \mathbb{C}^n$ with $n \geq 2$? In one variable, complex differentiability, the Cauchy–Riemann equations, and [power series](/page/Power%20Series) expansion are all equivalent — but we need to decide which property to take as the *definition* before checking that the others follow. The Wirtinger derivatives give a clean algebraic formulation of the Cauchy–Riemann equations in $n$ variables, and we start there. The Wirtinger derivatives are the fundamental differential operators of several complex variables. For each $j = 1, \dots, n$, define: [definition: Wirtinger Derivatives] Let $z_j = x_j + i y_j$ be coordinates on $\mathbb{C}^n$. The **Wirtinger derivatives** with respect to $z_j$ and $\bar{z}_j$ are the first-order differential operators \begin{align*} \partial_{z_j} &= \frac{1}{2}\left(\frac{\partial}{\partial x_j} - i\frac{\partial}{\partial y_j}\right), \qquad \partial_{\bar{z}_j} = \frac{1}{2}\left(\frac{\partial}{\partial x_j} + i\frac{\partial}{\partial y_j}\right). \end{align*} [/definition] These operators satisfy $\partial_{z_j} z_k = \delta_{jk}$, $\partial_{z_j} \bar{z}_k = 0$, $\partial_{\bar{z}_j} z_k = 0$, and $\partial_{\bar{z}_j} \bar{z}_k = \delta_{jk}$, exactly as formal differentiation of polynomials in $z_j, \bar{z}_j$ would suggest. The Cauchy–Riemann equations in several variables simply assert that each $\partial_{\bar{z}_j}$ annihilates $f$. The central object of the course is the class of functions that can be differentiated complex-linearly in all variables. Ordinary real differentiability is too weak for this role: a $C^1$ function may vary independently in the $z_j$ and $\bar z_j$ directions, while complex analysis only allows the $z_j$-dependence. The definition below names the functions that remove the anti-holomorphic directions and form the algebra used throughout the rest of the notes. [definition: Holomorphic Function] Let $\Omega \subset \mathbb{C}^n$ be open and $f: \Omega \to \mathbb{C}$ be a $C^1$ function (continuously real-differentiable). We say $f$ is **holomorphic** on $\Omega$ if \begin{align*} \partial_{\bar{z}_j} f &= 0 \quad \text{on } \Omega, \quad j = 1, \dots, n. \end{align*} The space of holomorphic functions on $\Omega$ is denoted $\mathcal{O}(\Omega)$. [/definition] Equivalently, $f \in C^1(\Omega)$ is holomorphic if and only if it is complex differentiable in each variable separately when the other variables are held fixed. When $n = 1$ this reduces to the usual Cauchy–Riemann condition. The class $\mathcal{O}(\Omega)$ is a $\mathbb{C}$-algebra closed under composition and locally uniform limits. ### Power Series in Several Variables In one complex variable, [power series](/page/Power%20Series) converge on discs. What is the right domain of convergence in $n$ variables? A ball $\{|z - a| < r\}$ in $\mathbb{C}^n$ seems natural, but the Cauchy formula integrates over products of circles in each coordinate — not over a sphere. The natural domain is therefore a *product* of discs, one in each variable, and these products are called polydiscs. Why do we need polydiscs rather than balls? Because when we iterate the one-variable Cauchy formula, we integrate each variable independently, and the region where that iteration converges is exactly a polydisc. In one complex variable, a [holomorphic function](/page/Holomorphic%20Function) is locally represented by a convergent [power series](/page/Power%20Series). The same holds in several variables, with the polydisc playing the role of the disc. [definition: Polydisc] For $a = (a_1, \dots, a_n) \in \mathbb{C}^n$ and $r = (r_1, \dots, r_n)$ with each $r_j > 0$, the **polydisc** centred at $a$ with polyradius $r$ is \begin{align*} \mathbb{D}^n(a, r) &:= \{z \in \mathbb{C}^n : |z_j - a_j| < r_j,\ j = 1, \dots, n\}. \end{align*} The **unit polydisc** is $\mathbb{D}^n := \mathbb{D}^n(0, (1,\dots,1))$, i.e., the Cartesian product of $n$ open unit discs. [/definition] A **[power series](/page/Power%20Series)** in $n$ variables centred at $a$ is a formal sum \begin{align*} \sum_{\alpha \in \mathbb{N}^n} c_\alpha (z - a)^\alpha \end{align*} where $\alpha = (\alpha_1, \dots, \alpha_n)$ is a multi-index, $(z-a)^\alpha = (z_1 - a_1)^{\alpha_1} \cdots (z_n - a_n)^{\alpha_n}$, and $c_\alpha \in \mathbb{C}$. Such a series converges absolutely and uniformly on any polydisc $\mathbb{D}^n(a, \rho)$ with $\rho_j < r_j$ for all $j$, provided it converges at some point $b$ with $|b_j - a_j| = r_j$ for each $j$. A function representable by a convergent [power series](/page/Power%20Series) near each point of $\Omega$ is called **analytic**. ### The Cauchy Integral Formula on Polydiscs Given a [holomorphic function](/page/Holomorphic%20Function) on a polydisc, can we recover its values from its values on the boundary? In one variable, the Cauchy formula integrates over the full boundary circle. In $n$ variables, the boundary of a polydisc $\mathbb{D}^n(a,r)$ is a real $(2n-1)$-dimensional manifold, but the Cauchy formula does *not* integrate over this full boundary — it integrates only over the **distinguished boundary**, the $n$-dimensional torus $\{|\zeta_j - a_j| = r_j : j = 1,\dots,n\}$. This is a striking difference: much less boundary information is needed, which has deep consequences for extension phenomena explored in Chapter 2. The [Cauchy integral formula](/theorems/345) in one variable extends to polydiscs by iterating the one-variable formula in each coordinate. [quotetheorem:3398] [citeproof:3398] The formula explains why polydiscs are the natural local domains in this theory: each coordinate contributes its own one-variable Cauchy kernel, and the resulting integral sees the product of boundary circles rather than the full topological boundary. It also shows that holomorphic functions are locally controlled by boundary data in a way strong enough to recover all derivatives at the centre. Consequently, the coefficients in the local [power series](/page/Power%20Series) expansion are determined by the integral formula \begin{align*} c_\alpha &= \frac{1}{\alpha!} \partial^\alpha f(a) = \frac{1}{(2\pi i)^n} \oint \cdots \oint \frac{f(\zeta)}{(\zeta - a)^{\alpha + \mathbf{1}}}\, d\zeta_1 \cdots d\zeta_n, \end{align*} where $\mathbf{1} = (1, \dots, 1)$ and the integration is over the distinguished boundary $\{|\zeta_j - a_j| = r_j\}$ of the polydisc. ### Osgood's Lemma and the Equivalence of Definitions In one variable, a function that is complex differentiable at every point is automatically smooth and analytic — no separate assumption of continuity is needed (this is a theorem of Goursat). In several variables, there is an extra logical step: one must show that holomorphicity on coordinate slices locks together into holomorphicity on polydiscs. The first theorem below gives an elementary bridge under a local boundedness hypothesis. [definition: Separate Holomorphicity] A function $f: \Omega \to \mathbb{C}$ on an [open set](/page/Open%20Set) $\Omega \subset \mathbb{C}^n$ is **separately holomorphic** if for each $j \in \{1, \dots, n\}$ and each fixed $(z_1, \dots, \hat{z}_j, \dots, z_n) \in \mathbb{C}^{n-1}$ (where $\hat{z}_j$ means $z_j$ is omitted), the map $z_j \mapsto f(z_1, \dots, z_n)$ is holomorphic on the appropriate open subset of $\mathbb{C}$. [/definition] The point is not that separately holomorphic functions on polydomains produce elementary counterexamples; Hartogs's theorem will show that they do not. Rather, the point is methodological: separate one-variable Cauchy estimates must first be upgraded to uniform estimates on neighbourhoods. Local boundedness is the clean hypothesis that supplies this upgrade. Once those estimates are available, the polydisc Cauchy formula becomes a structural tool rather than only an integral representation. [quotetheorem:3379] [citeproof:3379] Conceptually, local boundedness is the estimate that lets the one-variable information in each coordinate lock together into joint continuity, the polydisc Cauchy formula, and a local [power series](/page/Power%20Series) expansion. It turns a family of slice-wise holomorphic functions into one genuinely several-variable object. This is the first place where polydiscs matter: the Cauchy integral can be applied in each coordinate with uniform control on a product neighbourhood, producing coefficients indexed by multi-indices rather than by a single integer. The next theorem packages the resulting equivalence of definitions, since the rest of the course moves freely among whichever formulation is most convenient. [quotetheorem:3399] [citeproof:3399] Notice what the equivalence theorem gives us beyond the three conditions themselves. First, holomorphic functions are automatically $C^\infty$ — unlike the real case, where $C^1$ and analytic are genuinely different classes. This follows from the [power series](/page/Power%20Series) representation, which can be differentiated term by term to any order. Second, the hypothesis of local boundedness in condition (3) is not as restrictive as it sounds: Hartogs's theorem, proved in the next section, shows it can be dropped entirely. The boundedness assumption in [Osgood's lemma](/theorems/3379) is therefore an artifact of the elementary proof strategy, not an essential feature of the theory. [remark: Smoothness from Holomorphicity] Unlike the real case, where $C^1$ and analytic are genuinely different classes, holomorphic functions are automatically $C^\infty$ (and in fact real-analytic). This follows from the [power series](/page/Power%20Series) representation, which can be differentiated term by term to any order. [/remark] ## Separate Holomorphicity and Hartogs's Theorem The equivalence in [Osgood's lemma](/theorems/3379) required local boundedness as a hypothesis. A natural and important question is whether local boundedness can be dropped entirely — whether every separately [holomorphic function](/page/Holomorphic%20Function) is jointly holomorphic. This is [Hartogs's theorem on separate holomorphicity](/theorems/3400), and the answer is yes, even without any boundedness assumption. ### Towards Hartogs: Why Separate Implies Joint The strategy is to use a Baire category argument to pass from separate holomorphicity to local boundedness on a dense [open set](/page/Open%20Set), and then to invoke [Osgood's lemma](/theorems/3379). The [Baire category theorem](/page/Baire%20Category%20Theorem) is the key tool: a [complete metric space](/page/Complete%20Metric%20Space) (such as $\mathbb{C}^{n-1}$ or a polydisc) cannot be written as a countable union of nowhere-dense closed sets. [quotetheorem:3400] [citeproof:3400] Hartogs's theorem is remarkable because it has no analogue in real analysis: there exist functions $\mathbb{R}^2 \to \mathbb{R}$ that are smooth in each variable separately but are discontinuous (the standard example is $(x_1 x_2)/(x_1^2 + x_2^2)$ with value $0$ at the origin). Complex analyticity imposes such rigid constraints that separate and joint regularity coincide. [example: Separate Holomorphicity in Practice] Hartogs's theorem is most useful when joint holomorphicity is *not* apparent from the formula. Here is a structural example that shows the theorem doing real work. Let $\Omega = \mathbb{C}^2 \setminus \{z_1^2 + z_2^2 = 0\}$ and define \begin{align*} f(z_1, z_2) &= \frac{z_1}{z_1^2 + z_2^2} \end{align*} on this domain. For each fixed $z_2 \neq 0$, the map $z_1 \mapsto z_1/(z_1^2 + z_2^2)$ is a rational function of $z_1$ with poles at $z_1 = \pm iz_2$ — holomorphic wherever it is defined. For each fixed $z_1$, the map $z_2 \mapsto z_1/(z_1^2 + z_2^2)$ is similarly a rational function holomorphic away from $z_2 = \pm iz_1$. So $f$ is separately holomorphic on $\Omega$. Is $f$ jointly holomorphic? The formula involves no antiholomorphic terms, so we expect yes, but the Cauchy–Riemann check requires verifying that no hidden $\bar{z}_j$ enters through the denominator. The direct verification is to write $z_j = x_j + iy_j$ and check $\partial_{\bar{z}_1} f = \partial_{\bar{z}_2} f = 0$ explicitly. Instead, Hartogs's theorem gives this immediately: $f$ is separately holomorphic, and on any compact subset of $\Omega$ it is bounded (being a continuous function on a compact set away from the singular locus), so [Osgood's lemma](/theorems/3379) applies and $f \in \mathcal{O}(\Omega)$. The zero set of $f$ is $\{z_1 = 0\} \cap \Omega = \{z_1 = 0,\, z_2 \neq 0\}$, which is a complex hypersurface minus a finite set — consistent with the theorem that no zero is isolated for $n \geq 2$. [/example] ## Zero Sets and the Identity Principle One of the most striking differences between one and several complex variables concerns the structure of zero sets. In one variable, the zeros of a non-trivial [holomorphic function](/page/Holomorphic%20Function) are isolated — they form a discrete set. In several variables, isolated zeros are impossible for $n \geq 2$, and zero sets are far richer geometric objects. ### The Identity Principle in Several Variables In one variable, agreement on a set with an accumulation point forces two holomorphic functions to agree everywhere on a connected domain. In several variables, that accumulation-point formulation is too strong to survive unchanged: lower-dimensional complex sets can have many accumulation points without forcing global equality. The safe several-variable form is this: if $\Omega \subset \mathbb{C}^n$ is connected and two functions $f,g \in \mathcal{O}(\Omega)$ agree on a nonempty open subset of $\Omega$, then $f=g$ on all of $\Omega$. Equivalently, if a holomorphic function vanishes on a nonempty open subset of a connected domain, then it vanishes identically. The proof follows the same continuation idea as in one variable: the set of points near which $f-g$ is identically zero is both open and closed in $\Omega$. The difference is visible in examples. The function $f(z_1,z_2)=z_1$ vanishes on the complex hypersurface $\{z_1=0\}$, which has accumulation points everywhere along it, but $f$ is not identically zero. Thus in several variables, "many zeros" is not the same as "zeros on an open set." It is the *consequence* about zero sets that changes drastically when $n \geq 2$. ### Why Isolated Zeros Are Impossible for $n \geq 2$ The identity principle only says that a holomorphic function cannot vanish on an open set without vanishing everywhere. In several variables there is a stronger geometric restriction: zeros of a nonzero holomorphic function must spread in complex directions rather than sit as isolated points. The next result makes this propagation precise and explains why singularity theory in higher dimension cannot be modelled on isolated one-variable poles. [quotetheorem:3380] [citeproof:3380] [explanation: Why Codimension Matters] The intuition here is topological. In $\mathbb{C}^1 \cong \mathbb{R}^2$, a point has real codimension $2$, which is the same as the codimension of a zero of a [holomorphic function](/page/Holomorphic%20Function) (a discrete point). Removing such a point disconnects small circles around it, giving the residue and [winding number](/page/Winding%20Number) their meaning. In $\mathbb{C}^2 \cong \mathbb{R}^4$, a zero set of a [holomorphic function](/page/Holomorphic%20Function) is a complex curve — a real $2$-dimensional surface. Such a surface has real codimension $2$ in $\mathbb{R}^4$, meaning it is "non-negligible" in exactly the right sense for function theory. But a single point in $\mathbb{R}^4$ has real codimension $4$, which is too large: a loop in $\mathbb{R}^4 \setminus \{0\}$ can always be pushed off a single point (the complement of a point in $\mathbb{R}^4$ is simply connected), so there is no [winding number](/page/Winding%20Number) detecting whether $f$ vanishes at the point. The analytic zero set must propagate at least along a complex curve. This codimension count underlies many deeper phenomena in the course. The [Hartogs extension theorem](/theorems/3401) shows forced extension across Hartogs figures, while the later Hartogs--Bochner theorem gives a compact-set version under a connected-complement hypothesis. For analytic singularity sets, complex codimension at least $2$ is the threshold at which holomorphic functions cannot support genuine poles there. [/explanation] ### A Worked Example: The Function $f(z_1, z_2) = 1/(z_1^2 + z_2^2)$ [example: Singularities in Several Variables] Consider the function \begin{align*} f(z_1, z_2) &= \frac{1}{z_1^2 + z_2^2} \end{align*} defined on $\mathbb{C}^2 \setminus \{z_1^2 + z_2^2 = 0\}$. The singular set is $V = \{z_1^2 + z_2^2 = 0\} = \{(z_1, z_2) : (z_1 + iz_2)(z_1 - iz_2) = 0\}$, which is the union of two complex hyperplanes $\{z_1 = \pm i z_2\}$. This is a complex curve in $\mathbb{C}^2$ — a $2$-real-dimensional set — not an isolated singularity. Even the origin $(0,0) \in V$ is not an isolated point of $V$: it lies on both lines $z_1 = iz_2$ and $z_1 = -iz_2$. Compare with the one-variable function $g(z) = 1/(z^2 + 1) = 1/((z+i)(z-i))$, which has isolated poles at $z = \pm i$. The transition from $n=1$ to $n=2$ replaces isolated pole points with genuine complex-analytic hypersurfaces. Notice also that $f$ is separately meromorphic: for fixed $z_2$ with $z_2 \neq 0$, the function $z_1 \mapsto 1/(z_1^2 + z_2^2)$ is meromorphic with poles at $z_1 = \pm iz_2$; for fixed $z_2 = 0$, the function $z_1 \mapsto 1/z_1^2$ is meromorphic. The singular set has no isolated components; it is a reducible complex curve, namely the union of the two hyperplanes displayed above. [/example] ### The Weierstrass Preparation Theorem The preceding discussion says that zero sets cannot usually be treated as collections of isolated points. To study them locally, we need a substitute for the one-variable factorisation of a function with a zero as $(z - a)^k \cdot (\text{unit})$. The [Weierstrass preparation theorem](/theorems/3381) provides that substitute by singling out one coordinate and replacing the function, up to a holomorphic unit, by a polynomial in that coordinate with holomorphic coefficients in the others. [quotetheorem:3381] [citeproof:3381] The theorem should be read as a local normal form. Near a non-flat zero, a several-variable holomorphic function behaves like a one-variable polynomial in a distinguished coordinate, multiplied by an invertible holomorphic factor. This is what lets zero sets in several variables be studied by mixing analytic geometry with one-variable polynomial algebra. The hypothesis that $f(0,\dots,0,z_n) \not\equiv 0$ is essential: it ensures $f$ has a well-defined finite order of vanishing $k$ in the $z_n$-direction. Without this, the zero set of $f$ would contain the entire $z_n$-axis near the origin, and no Weierstrass polynomial factorisation exists. The hypothesis is generically satisfied after a linear change of coordinates: for any $f \in \mathcal{O}(U)$ with $f(0) = 0$ and $f \not\equiv 0$, there exists a linear automorphism $L$ of $\mathbb{C}^n$ such that $(f \circ L)(0,\dots,0,z_n) \not\equiv 0$ — in other words, almost every direction serves as a valid distinguished variable, and one can always arrange the hypothesis by a preliminary coordinate rotation. [remark: Consequences of the Preparation Theorem] The [Weierstrass preparation theorem](/theorems/3381) implies that zero sets of holomorphic functions are locally given by the zero set of a polynomial in the last variable with holomorphic coefficients. This shows that zero sets are complex-analytic hypersurfaces (real codimension $2$), confirming our earlier discussion. It also underlies the algebraic structure of $\mathcal{O}_{0,\mathbb{C}^n}$ (the local ring of germs at the origin), which is a Noetherian [unique factorisation domain](/page/Unique%20Factorisation%20Domain) — a key ingredient in the theory of coherent analytic sheaves studied later in the course. [/remark] ## Comparing One and Several Variables: A Summary The results of this chapter reveal a sharp contrast between function theory in one and several complex variables. [explanation: One vs. Several Variables] In one complex variable ($n = 1$), three pillars of the theory are: **Isolated singularities.** A [holomorphic function](/page/Holomorphic%20Function) can have an isolated singularity at a point $z_0$; this singularity is classified as removable, a pole, or essential (Riemann, Weierstrass–Casorati). Isolated singularities are a rich source of local invariants (residues, winding numbers). **Isolated zeros.** If $f \in \mathcal{O}(\Omega)$ is not identically zero, its zero set is discrete — the zeros are isolated and have finite order. **Separate = joint holomorphicity.** In one variable this is a vacuous condition (there is only one variable). **Riemann mapping theorem.** Every simply connected proper open subset of $\mathbb{C}$ is biholomorphic to the unit disc. In several complex variables ($n \geq 2$), each of these fails or transforms fundamentally: **No isolated singularities.** Analytic singularity sets of complex codimension $\geq 2$ are removable in the Hartogs sense, and compact holes with connected complement are filled by the Hartogs--Bochner theorem. There is no analogue of an isolated essential singularity. **No isolated zeros.** By the theorem above, zero sets of holomorphic functions in $\mathbb{C}^n$ are complex hypersurfaces — real-codimension-2 manifolds — never isolated points. **Separate holomorphicity is joint.** Hartogs's theorem, proved in this chapter, shows that separate holomorphicity already forces joint holomorphicity. **No Riemann mapping theorem.** Poincaré's theorem (Chapter 10) shows the unit ball $\mathbb{B}^2$ and the bidisc $\mathbb{D}^2$ are biholomorphically inequivalent, so there is no classification of simply connected domains by a single invariant. These differences are not incidental — they reflect deep geometric constraints on complex manifolds in higher dimensions. Chapter 2 studies forced extension across Hartogs figures; Chapter 3 introduces domains of holomorphy and the [Cartan–Thullen theorem](/theorems/3385); Chapters 4–5 develop plurisubharmonic functions and pseudoconvexity; Chapters 6–7 provide the $\bar\partial$-machinery for global problems; Chapter 8 solves the Levi problem; Chapter 9 makes the theory explicit for Reinhardt domains; and Chapter 10 establishes that the biholomorphic classification of domains in $\mathbb{C}^n$ is genuinely richer than in one variable. [/explanation] --- The [Hartogs extension theorem](/theorems/3401) reveals that holomorphic functions in several variables exhibit rigid extension behavior absent in one dimension—they are forced to extend across certain gaps in their domain. This phenomenon is not pathological but rather encodes which domains can support non-trivial holomorphic functions, leading to the central question: what geometric properties characterize domains where extension cannot occur? # 2. The Hartogs Extension Phenomenon The most startling discovery of several complex variables is that holomorphic functions can be forced to extend. In one variable, a function holomorphic on an annulus need not extend across the inner circle — a simple [Laurent series](/page/Laurent%20Series) with negative powers makes this explicit. In several variables, the situation reverses: for $n \geq 2$, certain "holes" in a domain are automatically filled in, not because of special properties of the function, but because of the topology of the domain itself. This chapter studies this extension phenomenon in depth, identifying the precise geometric conditions under which extension is forced, and drawing the first sharp consequences: the failure of isolated essential singularities and the inapplicability of Mittag-Leffler without global hypotheses. ## Hartogs Figures and the Extension Theorem The basic building block for forced extension is the *Hartogs figure*. To understand why it forces extension before stating the theorem, consider the one-variable analogy. On the punctured disc $\mathbb{D} \setminus \{0\}$, a [holomorphic function](/page/Holomorphic%20Function) with bounded modulus extends across $0$ by Riemann's theorem; but one with a Laurent expansion $\sum_{k=-\infty}^{-1} a_k z^k$ does not. The key feature is that a point in $\mathbb{C}$ is a divisor-type singular set: functions such as $1/z$ can have genuine poles there. In several variables, isolated missing points have complex codimension at least $2$, and Hartogs-type phenomena make them removable for holomorphic functions. In $\mathbb{C}^2$, the analogous configuration is a domain with a "bichromatic" complement: the part of the polydisc that one removes is not a complex hypersurface but something geometrically larger, which forces every [holomorphic function](/page/Holomorphic%20Function) to extend. [definition: Hartogs Figure] Fix $n \geq 2$ and $0 < \varepsilon < 1$. The *Hartogs figure* (or *Hartogs domain*) $H(\varepsilon) \subset \mathbb{C}^n$ is the [open set](/page/Open%20Set) \begin{align*} H(\varepsilon) = \bigl\{z \in \mathbb{D}^n : |z_n| > 1 - \varepsilon\bigr\} \cup \bigl\{z \in \mathbb{D}^n : |z_j| < \varepsilon \text{ for } j = 1, \ldots, n-1\bigr\}. \end{align*} This is the union of a thin outer shell $\{1 - \varepsilon < |z_n| < 1\}$ in the polydisc with a thin inner polydisc $\{|z_1|, \ldots, |z_{n-1}| < \varepsilon, |z_n| < 1\}$. [/definition] Equivalently, $H(\varepsilon)$ is a domain obtained by keeping both an outer shell in the last coordinate and an inner cylinder in the first $n-1$ coordinates. The missing region is not a one-variable annular hole: the two retained pieces overlap in a way that gives holomorphic functions access to all coordinate directions. This geometry sets up the first genuine surprise of the chapter. In one variable, an annulus supports Laurent series with negative powers, so the inner hole can carry a singularity. In a Hartogs figure, the extra variables force those would-be singular terms to disappear. The theorem below states this forced filling-in precisely. [quotetheorem:3401] [citeproof:3401] The important takeaway is that the missing core of the Hartogs figure is not a genuine singular barrier. Holomorphic data on the outer shell and inner cylinder are forced to continue across the whole polydisc, so several-variable holomorphy has a built-in extension mechanism absent from one-variable annuli. The theorem should be interpreted as a rigidity statement about domains, not as a removable-singularity theorem for one special function. Once a Hartogs figure sits inside a larger polydisc, the surrounding geometry dictates the analytic continuation region. This is why Hartogs figures become the local model for later extension theorems: they turn a geometric hole into forced continuation. [remark: The dimension hypothesis is essential] The argument fails entirely for $n = 1$. If $0<\varepsilon<1/2$, the one-variable analogue becomes $H(\varepsilon) = \{1-\varepsilon < |z| < 1\} \cup \{|z| < \varepsilon\}$, a disjoint union of an annulus and a small disc. A function can be defined to be $1$ on the small disc and $0$ on the annulus, with no possible holomorphic extension. The higher-dimensional connected overlap — arising from the independent coordinate conditions in the definition of $H(\varepsilon)$ — is what makes the Hartogs extension phenomenon genuinely several-variable. [/remark] [example: Extension across the core] Let $n = 2$ and $\varepsilon = 1/2$. Consider a [holomorphic function](/page/Holomorphic%20Function) $f$ on the Hartogs figure \begin{align*} H(1/2) = \bigl\{(z_1, z_2) \in \mathbb{D}^2 : |z_2| > 1/2\bigr\} \cup \bigl\{(z_1, z_2) \in \mathbb{D}^2 : |z_1| < 1/2\bigr\}. \end{align*} The theorem guarantees that $f$ extends to all of $\mathbb{D}^2$. The important point in this example is not the formula for the extension, but the geometry of the domain: the outer shell and inner cylinder overlap enough to force the missing middle region to be filled in analytically. [/example] ## The Hartogs–Bochner Theorem and Filling in Singularity Sets The [Hartogs extension theorem](/theorems/3401) for Hartogs figures is the local prototype of a global phenomenon: holomorphic functions extend across compact singularity sets, provided the complement is connected. The precise result in this direction is the [Hartogs–Bochner theorem](/theorems/3382). [quotetheorem:3382] [citeproof:3382] The theorem globalises the local Hartogs phenomenon: compact holes in several variables are often invisible to holomorphic functions. The connectedness of $\Omega \setminus K$ is the condition that prevents different components from carrying incompatible function values. For tube domains $\Omega = U + i\mathbb{R}^n$, the same filling principle can be interpreted through Fourier analysis in the imaginary directions. The geometric picture the [Hartogs–Bochner theorem](/theorems/3382) conveys is one of *filling in*. A compact subset $K$ of a domain $\Omega$ acts as an obstruction that holomorphic functions cannot detect: they automatically extend across it. Holomorphic functions in several variables cannot "see" compact holes inside a connected domain, in sharp contrast to the one-variable case. [explanation: Why the complement must be connected] The connectedness assumption on $\Omega \setminus K$ is necessary. If $K$ separates $\Omega$ into two components $\Omega_1$ and $\Omega_2$, a function that is $0$ on $\Omega_1$ and $1$ on $\Omega_2$ is holomorphic on both but has no consistent holomorphic extension to all of $\Omega$. In $\mathbb{C}^n$ with $n \geq 2$, a compact set $K \subset \Omega$ cannot disconnect $\Omega$ if the dimension of $K$ is at most $2n - 2$, by Alexander duality (or by simpler dimension-counting arguments). For example, a compact complex submanifold of complex dimension $k$ has real dimension $2k$, and it cannot separate $\mathbb{C}^n$ if $2k < 2n - 1$, i.e., $k \leq n - 2$. This is why codimension conditions on $K$ automatically guarantee the connectedness hypothesis in many applications. [/explanation] ## Removable Singularities in Several Variables [Riemann's removable singularity theorem](/theorems/3356) in one variable states that a [holomorphic function](/page/Holomorphic%20Function) on a punctured disc that is bounded extends across the puncture. How much of this carries over to several variables? In fact, far more holds: singularity sets of codimension $\geq 2$ are removable without any boundedness hypothesis. To state this precisely, recall from Chapter 1 that zero sets of holomorphic functions in $\mathbb{C}^n$ are complex hypersurfaces of real codimension $2$ — they cannot be isolated points when $n \geq 2$. The removability theorem for analytic sets reflects the same phenomenon. [definition: Analytic Set] A subset $A \subset \Omega$ of an open domain $\Omega \subset \mathbb{C}^n$ is an *analytic set* (or *analytic variety*) if for every point $p \in \Omega$ there is a neighbourhood $U$ of $p$ and holomorphic functions $g_1, \ldots, g_k \in \mathcal{O}(U)$ such that $A \cap U = \{z \in U : g_1(z) = \cdots = g_k(z) = 0\}$. [/definition] The *complex codimension* of an analytic set $A \subset \Omega$ is the difference $n - \dim_{\mathbb{C}} A$, where $\dim_{\mathbb{C}} A$ is the maximum complex dimension of the smooth part of $A$. An analytic set of codimension $\geq 2$ in $\mathbb{C}^n$ is a set on which at least two independent holomorphic equations are imposed — think of the origin in $\mathbb{C}^2$, which has codimension $2$. This codimension threshold is exactly where the extension phenomenon becomes stronger than the one-variable removable singularity theorem. The question is whether a holomorphic function defined away from such a thin analytic set has any freedom to develop a genuine singularity there, or whether the surrounding values force a unique holomorphic continuation. [quotetheorem:3383] [citeproof:3383] [example: Extension across the origin in $\mathbb{C}^2$] The simplest illustration is the origin $\{0\} \subset \mathbb{C}^2$, which is an analytic set of codimension $2$ (it is the zero set of $z_1$ and $z_2$ simultaneously). Every function $f \in \mathcal{O}(\mathbb{C}^2 \setminus \{0\})$ that is holomorphic off the origin extends to an entire function on $\mathbb{C}^2$. This stands in complete contrast to the one-variable situation: $f(z) = 1/z$ is holomorphic on $\mathbb{C} \setminus \{0\}$ with no holomorphic extension to $\mathbb{C}$. In two variables, no such function can exist — there is no room for an isolated singularity because the codimension is too small relative to the ambient dimension. To see why the theorem does not apply to *all* singularity sets, consider $f(z_1, z_2) = 1/(z_1^2 + z_2^2)$. This function is holomorphic away from the variety $\{z_1^2 + z_2^2 = 0\}$, which is a complex hypersurface (codimension $1$) passing through the origin — not just the origin itself. The theorem does not apply to codimension-$1$ sets, and indeed $1/(z_1^2 + z_2^2)$ does not extend across its polar set. [/example] ## Consequences: No Isolated Essential Singularities, No Mittag-Leffler The Hartogs extension phenomenon forces a fundamental restructuring of the singularity theory that one-variable analysts take for granted. ### Absence of Isolated Essential Singularities In one complex variable, essential singularities are a rich source of interesting behaviour: Picard's theorem, the [Casorati–Weierstrass theorem](/theorems/3355), and the classification of singularities all hinge on the existence of isolated essential singularities. What happens to this classification in several variables? [quotetheorem:3384] [citeproof:3384] This means there are no isolated poles, no isolated essential singularities, and no isolated removable singularities (of the interesting kind). The only singularities that can occur are along complex hypersurfaces — sets of codimension $1$ — which are genuine obstructions. Poles must occur along divisors: if $f$ has a "singularity" at $z$, then $z$ lies on a holomorphic hypersurface where $f$ blows up. [remark: The Hartogs triangle as a genuine boundary] Not every domain in $\mathbb{C}^n$ has the extension property. The *Hartogs triangle* $\Delta = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| < |z_2| < 1\}$ is an example of a domain from which holomorphic functions do not automatically extend to any larger domain. The function $f(z_1, z_2) = z_1/z_2$, holomorphic on $\Delta$, does not extend across the origin $(0,0)$ in a consistent way; the function $z_2/(z_2 - z_1)$ provides a [holomorphic function](/page/Holomorphic%20Function) that cannot be continued past the diagonal boundary $\{|z_1| = |z_2|\}$. In fact the Hartogs triangle is a domain of holomorphy — as Chapter 9 confirms via logarithmic convexity of its log-image — and it shows that extension phenomena are domain-specific, not universal. [/remark] ### Failure of the Mittag-Leffler Theorem Without Global Hypotheses Recall that the classical [Mittag-Leffler theorem](/theorems/3367) in one variable allows prescribing the singular parts at a discrete set of poles: given distinct points $a_1, a_2, \ldots$ in $\mathbb{C}$ and principal parts $P_k(z) = \sum_{j=1}^{m_k} c_{jk}/(z - a_k)^j$, there exists a [meromorphic function](/page/Meromorphic%20Function) on $\mathbb{C}$ with exactly these singularities. In several variables, the analogous construction fails without global domain hypotheses. The obstruction is not analytic but topological. [explanation: Why Mittag-Leffler needs a global hypothesis] The several-variable analogue of Mittag-Leffler is the Cousin I problem: one prescribes local meromorphic data whose differences are holomorphic on overlaps and asks whether a single global meromorphic function realizes those prescriptions. In one variable this is governed by the classical Mittag-Leffler theorem. In several variables the answer depends on the domain. Domains of holomorphy are the good setting; arbitrary domains can fail because holomorphic functions may be forced to extend to a larger region, leaving too little freedom to prescribe independent local singular data. [/explanation] [example: A Non-Stein Warning] The complement $\Omega=\mathbb{C}^2\setminus\{0\}$ is the basic warning example. It is not a domain of holomorphy: holomorphic functions on $\Omega$ extend across the missing point. This does not yet require the reader to know sheaf cohomology. The lesson is simpler: if every holomorphic function on a domain secretly belongs to a larger domain, then local meromorphic prescriptions on the smaller domain may be constrained by that larger extension. [/example] The Hartogs extension phenomenon thus creates a sharp division in the landscape of domains in $\mathbb{C}^n$. Domains from which holomorphic functions automatically extend are fundamentally different in character from domains of holomorphy — they lack the "boundary completeness" that makes complex analysis well-behaved globally. Identifying which domains are domains of holomorphy, and characterising them intrinsically via holomorphic convexity, is the central project of Chapter 3. --- A domain where holomorphic functions cannot be forced to extend is precisely a domain of holomorphy, and the [Hartogs extension theorem](/theorems/3401) shows these domains are rare and geometrically special. Understanding when a domain is (or is not) a domain of holomorphy requires both the extension criterion from Chapter 2 and tools to detect "holes" in the holomorphic hull—a task that demands new analytic machinery. # 3. Domains of Holomorphy The [Hartogs extension theorem](/theorems/3401) from the previous chapter revealed that certain domains in $\mathbb{C}^n$ are fundamentally "too small" — every [holomorphic function](/page/Holomorphic%20Function) on them extends to a strictly larger domain. This raises the opposite question: which domains are "maximal" in the sense that there exist holomorphic functions which genuinely need the whole domain? These are the domains of holomorphy, and characterising them is the central problem of this course. It turns out that being a domain of holomorphy is equivalent to a purely internal geometric condition — holomorphic convexity — a deep fact captured by the [Cartan–Thullen theorem](/theorems/3385). ## Domains of Holomorphy The difficulty encountered with Hartogs figures motivates a precise definition of those domains which do not suffer from forced extension. ### The Problem Hartogs Figures Reveal In one variable, every domain $\Omega \subset \mathbb{C}$ is a domain of holomorphy: given any boundary point $p \in \partial \Omega$, the function $f(z) = 1/(z - p)$ is holomorphic on $\Omega$ and blows up at $p$, making continuation across $p$ impossible. The reason this works is that $1/(z-p)$ has a pole precisely at $p$, a boundary point. In several variables, this argument breaks down because the zero set of a [holomorphic function](/page/Holomorphic%20Function) has complex codimension $1$, not codimension $n$. A boundary point of $\Omega \subset \mathbb{C}^n$ is a single point of real codimension $2n - 1 \geq 3$ in the boundary, but the singularity set of a [meromorphic function](/page/Meromorphic%20Function) is a complex hypersurface. If the boundary is curved, placing a polar singularity at a single point creates singularities that "wrap around" and force the function to extend. Hartogs figures show that for some domains, every [holomorphic function](/page/Holomorphic%20Function) on the domain extends to a larger domain regardless of how we try to build an obstruction. A domain of holomorphy is defined as one where this failure cannot happen: there exists a [holomorphic function](/page/Holomorphic%20Function) that genuinely cannot be continued past any boundary point. [definition: Domain of Holomorphy] A domain $\Omega \subset \mathbb{C}^n$ is a **domain of holomorphy** if there exists a function $f \in \mathcal{O}(\Omega)$ such that for every boundary point $p \in \partial \Omega$ and every open ball $B(p, r)$, the restriction $f|_{\Omega \cap B(p, r)}$ does not extend to a [holomorphic function](/page/Holomorphic%20Function) on $B(p, r)$. [/definition] Equivalently, $f$ cannot be holomorphically continued across any boundary point, even locally. Such $f$ is called a **[holomorphic function](/page/Holomorphic%20Function) of the domain**. [remark: Strength of the Condition] The definition requires a single function $f$ that cannot be continued across any point of the boundary simultaneously. One might have hoped that for each boundary point $p$ there could be a different function $f_p$ obstructing extension at $p$; the definition asks for a single $f$ that obstructs at all boundary points at once. This is a stronger requirement and makes domains of holomorphy harder to construct directly. [/remark] ## Examples and Non-Examples What kinds of domains are domains of holomorphy, and what kinds fail? The examples below span convex domains, classical symmetric spaces, and Hartogs figures, illustrating the range of behaviour and sharpening intuition for the [Cartan–Thullen theorem](/theorems/3385) to come. [example: Polydiscs Are Domains of Holomorphy] The polydisc $\mathbb{D}^n = \{z \in \mathbb{C}^n : |z_j| < 1 \text{ for all } j\}$ is a domain of holomorphy. One way to see this is structural: the polydisc is convex, and convex domains are holomorphically convex, hence domains of holomorphy by the Cartan–Thullen theorem. Coordinate functions still give useful local intuition. If $p \in \partial\mathbb{D}^n$ and $|p_j|=1$, then $1/(z_j-p_j)$ is holomorphic on the polydisc and blows up near that boundary point. These pointwise obstructions explain why the boundary is visible to holomorphic functions, while the theorem supplies the stronger single-function condition in the definition of a domain of holomorphy. [/example] The polydisc example is the simplest member of a larger class: convex domains. Convexity gives enough separating hyperplanes to control holomorphic hulls, even when no single coordinate direction is preferred. [example: Convex Domains Are Domains of Holomorphy] Every convex domain $\Omega \subset \mathbb{C}^n$ (viewed as an open convex subset of $\mathbb{R}^{2n}$) is a domain of holomorphy. The geometric reason is that convexity supplies supporting hyperplanes at boundary points, and these hyperplanes give enough complex-linear separation to keep holomorphic hulls inside the domain. Thus convexity implies holomorphic convexity, and Cartan–Thullen then identifies $\Omega$ as a domain of holomorphy. [/example] Biholomorphic equivalence provides a powerful shortcut: if a domain is known to be biholomorphic to one already verified as a domain of holomorphy, it inherits the property for free. This reduces the Siegel upper half-space to the case of the ball. [example: The Siegel Upper Half-Space] The Siegel upper half-space $\mathcal{H} = \{z \in \mathbb{C}^n : \operatorname{Im}(z_n) > |z_1|^2 + \cdots + |z_{n-1}|^2\}$ is a domain of holomorphy. It is biholomorphic to the unit ball $\mathbb{B}^n$, which is a convex domain. Since biholomorphic maps preserve the property of being a domain of holomorphy (a biholomorphism pulls back a witness function), $\mathcal{H}$ is a domain of holomorphy. [/example] [example: Hartogs Figures Are Not Domains of Holomorphy] The Hartogs figure $H = \{z \in \mathbb{D}^2 : |z_1| > 1/2\} \cup \{z \in \mathbb{D}^2 : |z_2| < 1/2\} \subset \mathbb{C}^2$ is not a domain of holomorphy. As proved in the previous chapter, every $f \in \mathcal{O}(H)$ extends to a [holomorphic function](/page/Holomorphic%20Function) on all of $\mathbb{D}^2$. So no function in $\mathcal{O}(H)$ can obstruct extension across the interior boundary points of $H$ (those points in $\partial H \cap \mathbb{D}^2$), and $H$ fails to be a domain of holomorphy. [/example] ## Holomorphic Convexity The definition of a domain of holomorphy is existential — it asserts the existence of a single obstructing function. This makes it difficult to verify directly. Holomorphic convexity reframes the same property in terms of the geometry of compact sets under the holomorphic functions on the domain, and is much more amenable to computation. In ordinary convexity, the convex hull of a compact set $K$ in $\mathbb{R}^n$ is characterised by linear functions: a point $x$ is in the convex hull of $K$ iff $\ell(x) \leq \sup_K \ell$ for every linear functional $\ell$. The holomorphic analogue replaces linear functionals by holomorphic functions and the absolute value by the modulus. [definition: Holomorphic Hull] Let $\Omega \subset \mathbb{C}^n$ be a domain and $K \subset \Omega$ a compact subset. The **holomorphic hull** of $K$ with respect to $\Omega$ is \begin{align*} \hat{K}_\Omega = \{z \in \Omega : |f(z)| \leq \sup_K |f| \text{ for all } f \in \mathcal{O}(\Omega)\}. \end{align*} [/definition] The holomorphic hull $\hat{K}_\Omega$ is always closed in $\Omega$ (it is an intersection of closed sets), and it contains $K$. The key question is whether it remains compactly contained in $\Omega$, i.e., whether $\hat{K}_\Omega \Subset \Omega$. The notation $A \Subset B$ means that $\overline{A}$ is compact and $\overline{A} \subset B$. [definition: Holomorphic Convexity] A domain $\Omega \subset \mathbb{C}^n$ is **holomorphically convex** if for every compact $K \Subset \Omega$, the holomorphic hull $\hat{K}_\Omega$ is also compactly contained in $\Omega$, i.e., $\hat{K}_\Omega \Subset \Omega$. [/definition] The geometric meaning of holomorphic convexity is that the class of holomorphic functions on $\Omega$ is "rich enough" to separate $K$ from the boundary: if a point $z$ is close to $\partial \Omega$ (not compactly separated from it), some $f \in \mathcal{O}(\Omega)$ must be large at $z$ compared to its values on $K$. [example: The Holomorphic Hull of the Distinguished Boundary of a Polydisc] Consider the torus $K = \{z \in \mathbb{C}^2 : |z_1| = |z_2| = 1\}$, the distinguished boundary of the polydisc $\mathbb{D}^2$, and compute $\hat{K}_{\mathbb{C}^2}$ — the hull with respect to all entire functions on $\mathbb{C}^2$. For any polynomial $f(z_1, z_2) = z_1^a z_2^b$ with $a, b \geq 0$, the maximum principle gives $\sup_K |f| = 1$. A point $z = (z_1, z_2)$ lies in $\hat{K}_{\mathbb{C}^2}$ only if $|z_1^a z_2^b| \leq 1$ for all $a, b \geq 0$, which requires $|z_1| \leq 1$ and $|z_2| \leq 1$. Conversely, for any $(z_1, z_2)$ with $|z_1|, |z_2| \leq 1$, the [maximum modulus principle](/page/Maximum%20Modulus%20Principle) applied to $z_1 \mapsto f(z_1, z_2)$ and $z_2 \mapsto f(z_1, z_2)$ shows that $|f(z_1, z_2)| \leq \sup_K |f|$ for any holomorphic $f$. Thus \begin{align*} \hat{K}_{\mathbb{C}^2} = \overline{\mathbb{D}^2} = \{(z_1, z_2) : |z_1| \leq 1, |z_2| \leq 1\}, \end{align*} the closed polydisc. This example is not a hull computation inside the open unit polydisc, because $K$ lies on its boundary. It is a computation in $\mathbb{C}^2$ showing the filling phenomenon: the distinguished torus has a hull with interior. If the ambient domain is a slightly enlarged polydisc, then this filled hull is compactly contained in that ambient domain. [/example] In contrast, when the ambient domain has a hole — such as an annulus — the hull cannot fill in to a solid region, because functions holomorphic on the annulus detect both inner and outer boundaries. The polyannulus illustrates this rigidity. [example: The Holomorphic Hull of a Polyannulus] Let $A = \{z \in \mathbb{C}^2 : 1/2 < |z_1| < 1, 1/2 < |z_2| < 1\}$ be a polyannulus, and let $K = \{z \in A : |z_1| = 3/4, |z_2| = 3/4\}$ be the product of two circles. The functions $z_1^k$ for $k \in \mathbb{Z}$ are holomorphic on $A$. For $z = (z_1, z_2) \in \hat{K}_A$, we need $|z_1|^k \leq (3/4)^k$ for all $k \geq 1$ (giving $|z_1| \leq 3/4$) and $|z_1|^{-k} \leq (3/4)^{-k}$ for all $k \geq 1$ (giving $|z_1| \geq 3/4$). Therefore $|z_1| = 3/4$, and similarly $|z_2| = 3/4$. So $\hat{K}_A = K$ itself. This reflects the rigidity of the polyannulus: the holomorphic hull of a product of circles within a polyannulus is precisely that product of circles, not a filled-in solid region. [/example] ## The Cartan–Thullen Theorem The central theorem of this chapter equates the existential condition of being a domain of holomorphy with the geometric condition of holomorphic convexity. This is a major change of perspective: instead of producing a particular holomorphic function that refuses to extend, one can test compact sets and their holomorphic hulls inside the domain. The theorem is the bridge between analytic non-extension and the intrinsic geometry of the domain. [quotetheorem:3385] [citeproof:3385] Cartan–Thullen turns a difficult existential definition into a geometric test. Instead of searching directly for one holomorphic function that fails to continue across every boundary point, one can study how compact sets expand under holomorphic hulls. If hulls always remain compactly inside the domain, then the domain has enough holomorphic functions to be maximal for analytic continuation. Cartan–Thullen says that hulls stay compactly inside the domain, but applications often need a sharper quantitative version: if a compact set sits a definite Euclidean distance from the boundary, can its holomorphic hull move any closer to the boundary than the original set did? This is the exact issue in Hartogs-type examples, where a hull approaching a missing boundary point signals forced analytic continuation. The next theorem packages holomorphic convexity into a boundary-distance test, giving a computable way to detect when hulls are starting to escape. This boundary-distance viewpoint also prepares the later link with the function $-\log d(\cdot,\partial\Omega)$. Before moving to plurisubharmonic functions, we need this intermediate result because it translates the abstract hull condition into a concrete equality of distances to the boundary. It should still be used as a characterisation under the theorem's hypotheses, not as a standalone shortcut divorced from holomorphic convexity. [quotetheorem:3402] [citeproof:3402] The [Cartan–Thullen theorem](/theorems/3385) is the first of three equivalent characterisations of domains of holomorphy that this course develops. The full equivalence, to be completed in Chapter 8, adds pseudoconvexity (defined precisely in Chapter 5) and the plurisubharmonic condition on the distance function (established in Chapter 4): $\Omega$ is a domain of holomorphy $\iff$ $\Omega$ is holomorphically convex $\iff$ $\Omega$ is pseudoconvex $\iff$ $-\log d(\cdot, \partial\Omega)$ is plurisubharmonic on $\Omega$. The implication involving pseudoconvexity — the [solution of the Levi problem](/theorems/3416) — requires the machinery of plurisubharmonic functions (Chapter 4), pseudoconvexity definitions (Chapter 5), and the $\bar\partial$-equation (Chapters 6–7). ## A Hull That Detects a Missing Point The next example computes a hull directly, without assuming the domain is already a domain of holomorphy. It illustrates why the punctured space fails holomorphic convexity. [example: Thin Annular Shell in $\mathbb{C}^2$] Let $\Omega = \mathbb{C}^2 \setminus \{0\}$ (the complement of the origin in $\mathbb{C}^2$) and let $K = \{z \in \mathbb{C}^2 : |z| = 1\}$ be the unit sphere. The boundary of $\Omega$ is $\partial \Omega = \{0\}$, so $d(K, \partial\Omega) = \inf_{|z|=1} |z - 0| = 1$. By Hartogs's theorem, every function holomorphic on $\mathbb{C}^2 \setminus \{0\}$ extends to an entire function on $\mathbb{C}^2$ (since $\{0\}$ is a compact analytic set of codimension $2$). The holomorphic hull of $K$ with respect to $\mathcal{O}(\mathbb{C}^2 \setminus \{0\}) = \mathcal{O}(\mathbb{C}^2)|_{\mathbb{C}^2 \setminus \{0\}}$ therefore equals $\hat{K}_{\mathbb{C}^2} \cap (\mathbb{C}^2 \setminus \{0\})$. The hull $\hat{K}_{\mathbb{C}^2}$ with respect to entire functions is $\overline{B}(0, 1)$ (the closed ball), by the [maximum modulus principle](/theorems/491) for entire functions. So \begin{align*} \hat{K}_\Omega = \overline{B}(0, 1) \setminus \{0\}, \end{align*} which is not compactly contained in $\Omega = \mathbb{C}^2 \setminus \{0\}$ (its closure contains the origin). This confirms that $\mathbb{C}^2 \setminus \{0\}$ is not holomorphically convex, consistent with it not being a domain of holomorphy. [/example] The thin annular shell example is not merely a curiosity: it explains, via holomorphic convexity, why Hartogs's removal-of-singularities theorem is tied to the failure of the punctured space to be a domain of holomorphy. The hull of $K$ reaches back toward the removed origin, so the domain cannot keep holomorphic hulls compactly inside itself. The following explanation makes this connection between geometry and extendability precise. [explanation: Why Holomorphic Convexity Is the Right Notion] Ordinary convexity in $\mathbb{R}^n$ can be characterised by linear functionals: $K$ is convex iff $K = \hat{K}$ where $\hat{K} = \{x : \ell(x) \leq \sup_K \ell \text{ for all linear } \ell\}$. A domain $\Omega$ is convex iff $K \Subset \Omega$ implies $\hat{K} \Subset \Omega$. Holomorphic convexity is the exact analogue with holomorphic functions replacing linear functionals. This analogy is not superficial: convex domains are holomorphically convex (and hence are domains of holomorphy), and the proof uses exactly the separation argument from convexity lifted to complex linear functionals. The strength of the [Cartan–Thullen theorem](/theorems/3385) is that it shows holomorphic convexity is not merely a sufficient condition for being a domain of holomorphy — it is also necessary. This is surprising because the definition of a domain of holomorphy asks for a single function with a very strong global property (non-extendability across every boundary point simultaneously), while holomorphic convexity is a condition on all compact subsets and all holomorphic functions. The theorem says these two perspectives are perfectly matched. The distance characterisation $d(\hat{K}_\Omega, \partial\Omega) = d(K, \partial\Omega)$ gives a computable test: to check whether a domain is a domain of holomorphy, estimate the holomorphic hull and check whether points near $\partial\Omega$ belong to it. If they do, the domain fails holomorphic convexity and therefore fails to be a domain of holomorphy. [/explanation] --- The holomorphic hull and extension properties are fundamentally linked to curvature conditions on the boundary, which suggests that smooth, convex-like functions should capture the right geometric notion. Plurisubharmonic functions—the several-variable analogue of subharmonic functions—are precisely the class that encodes these curvature constraints and connects boundary geometry to the existence of holomorphic functions. # 4. Plurisubharmonic Functions Plurisubharmonic functions are the natural class of functions adapted to complex analysis in several variables, playing the same role that subharmonic functions play in one variable. The central novelty is the restriction along complex lines: a function is plurisubharmonic if its restriction to every complex affine line is subharmonic, a condition far more stringent than ordinary subharmonicity when $n \geq 2$. This chapter develops the definition, the Levi form criterion for smooth functions, the main stability properties, and the key examples connecting plurisubharmonicity to holomorphic functions and domain geometry. These tools will underpin the notion of pseudoconvexity in Chapter 5, and the psh function $-\log d(\cdot, \partial\Omega)$ will be central to the [solution of the Levi problem](/theorems/3416) in Chapter 8. ## Subharmonic Functions in One Variable To motivate the several-variable definition, we first recall the one-variable theory that it generalises. Plurisubharmonicity is defined by requiring subharmonicity along every complex line, so we need a precise picture of what subharmonicity means in $\mathbb{C}$ before lifting the condition to $\mathbb{C}^n$. A function that is everywhere below the average of its surrounding values cannot have local maxima — this is the essential property of subharmonic functions. In the complex plane, "surrounding values" are taken along circles, and "average" is the mean over the circle. [definition: Subharmonic Function] Let $U \subset \mathbb{C}$ be open. A function $v: U \to [-\infty, \infty)$ is **subharmonic** if: 1. $v$ is upper semicontinuous, 2. $v \not\equiv -\infty$ on any connected component of $U$, 3. for every $z_0 \in U$ and every $r > 0$ with $\overline{B}(z_0, r) \subset U$, \begin{align*} v(z_0) \leq \frac{1}{2\pi} \int_0^{2\pi} v(z_0 + re^{i\theta})\, d\theta. \end{align*} [/definition] This is the **sub-mean-value property**: the value at any interior point is at most the average over any surrounding circle. When $v$ is smooth, this is equivalent to $\Delta v \geq 0$, where $\Delta = \partial^2/\partial x^2 + \partial^2/\partial y^2 = 4\,\partial_z \partial_{\bar z}$ is the Laplacian with Wirtinger operators $\partial_z = \frac{1}{2}(\partial_x - i\partial_y)$ and $\partial_{\bar z} = \frac{1}{2}(\partial_x + i\partial_y)$. The key examples are: (i) any harmonic function (which satisfies $\Delta v = 0$) is subharmonic; (ii) for any holomorphic $f: U \to \mathbb{C}$, the function $\log|f|$ is subharmonic — at non-zeros of $f$ one checks $\Delta \log|f| = 0$ by the Cauchy–Riemann equations, while at zeros the value $-\infty$ satisfies the sub-mean-value inequality automatically since the average over any surrounding circle is greater than $-\infty$. Subharmonic functions are closed under taking pointwise maxima and under decreasing limits. Both of these closure properties will carry over to the several-variable setting. ## The Definition of Plurisubharmonicity Moving to $\mathbb{C}^n$ for $n \geq 2$, we must decide what "subharmonic in several variables" should mean. The naive approach — requiring the sub-mean-value property for all real circles in $\mathbb{R}^{2n}$ — gives ordinary subharmonicity for the real Laplacian on $\mathbb{R}^{2n}$. This turns out to be the wrong notion for complex analysis. [motivation] ### Why complex lines, not real lines? In $\mathbb{C}^n$ with $n \geq 2$, the real dimension is $2n$ and one can test the sub-mean-value property along circles in $2n$ independent real directions. Ordinary subharmonicity tests all such directions equally. But holomorphic functions are constrained only with respect to complex-linear directions, not all real ones. A notion of "subharmonic" that interacts correctly with holomorphic maps must be invariant under holomorphic changes of coordinates. Ordinary real subharmonicity is not: if $F: \Omega' \to \Omega$ is holomorphic, the pullback $u \circ F$ of a real-subharmonic function need not be real-subharmonic. Restricting the sub-mean-value property to circles lying in complex affine lines gives a condition that is preserved under holomorphic pullback. This is the condition we adopt, and the resulting class — plurisubharmonic functions — is the correct complex-analytic analogue of subharmonic functions. [/motivation] To see concretely why ordinary subharmonicity is insufficient, consider $u(z_1, z_2) = \operatorname{Re}(z_1 \bar{z}_2)$ on $\mathbb{C}^2$. Writing $z_j = x_j + iy_j$, this is $u = x_1 x_2 + y_1 y_2$, which is harmonic on $\mathbb{R}^4$ and therefore subharmonic. Yet it cannot belong to a class compatible with holomorphic geometry: as we compute in the non-example below, its complex Hessian has eigenvalues $\pm 1/2$, meaning it decreases along certain complex directions. The class of plurisubharmonic functions correctly excludes this example by requiring non-negativity of the complex Hessian, not merely of the real Laplacian. [definition: Plurisubharmonic Function] Let $\Omega \subset \mathbb{C}^n$ be open. A function $u: \Omega \to [-\infty, \infty)$ is **plurisubharmonic** (abbreviated **psh**) if: 1. $u$ is upper semicontinuous, 2. $u \not\equiv -\infty$ on any connected component of $\Omega$, 3. for every $z \in \Omega$ and every $w \in \mathbb{C}^n$, the restriction $\zeta \mapsto u(z + \zeta w)$ is subharmonic on the [open set](/page/Open%20Set) $\{\zeta \in \mathbb{C} : z + \zeta w \in \Omega\}$. [/definition] Condition (3) requires that for each complex line $\{z + \zeta w : \zeta \in \mathbb{C}\} \cap \Omega$, the restriction of $u$ to that line is subharmonic in the complex parameter $\zeta$. In particular, for every $z \in \Omega$, $w \in \mathbb{C}^n$, and $r > 0$ with $\{z + \zeta w : |\zeta| \leq r\} \subset \Omega$, the sub-mean-value inequality reads \begin{align*} u(z) \leq \frac{1}{2\pi}\int_0^{2\pi} u(z + re^{i\theta} w)\, d\theta. \end{align*} We denote the class of plurisubharmonic functions on $\Omega$ by $\operatorname{PSH}(\Omega)$. [remark: Relation to Subharmonic Functions] Every psh function on $\Omega \subset \mathbb{C}^n$ is subharmonic for the real Laplacian $\Delta_{\mathbb{R}^{2n}}$. To see this, note that taking $w$ to range over the standard complex basis vectors $e_1, \ldots, e_n$ and computing the resulting Laplacians in each complex coordinate recovers $\Delta_{\mathbb{R}^{2n}} u \geq 0$ for smooth $u$ (the full real Laplacian is a sum of complex Hessian entries over the diagonal). The converse fails when $n \geq 2$: ordinary subharmonicity does not imply psh, as the non-example below demonstrates. When $n = 1$, psh and subharmonic coincide, since every direction in $\mathbb{C}^1$ is a complex direction. [/remark] ## The Levi Form Criterion How do you verify in practice that a given smooth function is plurisubharmonic? Checking the sub-mean-value inequality along every complex line is not a finite computation. For smooth functions, however, plurisubharmonicity reduces to a single matrix condition that can be checked by computing second partial derivatives. [definition: Levi Form] Let $\Omega \subset \mathbb{C}^n$ be open and $u \in C^2(\Omega)$ be real-valued. The **Levi form** (or **complex Hessian**) of $u$ at a point $z \in \Omega$ is the $n \times n$ Hermitian matrix \begin{align*} \mathcal{L}_u(z) = \left( \frac{\partial^2 u}{\partial z_j \partial \bar{z}_k}(z) \right)_{j,k=1}^n. \end{align*} Here $\partial/\partial z_j = \frac{1}{2}(\partial/\partial x_j - i\,\partial/\partial y_j)$ and $\partial/\partial \bar{z}_k = \frac{1}{2}(\partial/\partial x_k + i\,\partial/\partial y_k)$ are the Wirtinger derivatives, as defined in Chapter 1. [/definition] The matrix $\mathcal{L}_u(z)$ is Hermitian because $u$ is real-valued and smooth: $\overline{\partial^2 u/\partial z_j \partial \bar{z}_k} = \partial^2 u/\partial z_k \partial \bar{z}_j$ by Schwarz's theorem. The Levi form evaluates on a direction $\xi \in \mathbb{C}^n$ as the quadratic form $\sum_{j,k} (\partial^2 u/\partial z_j \partial \bar{z}_k)(z)\,\xi_j \overline{\xi_k}$. The point of introducing this matrix is that it compresses the line-by-line definition of plurisubharmonicity into a finite second-derivative test. Positivity of the quadratic form in every complex direction is exactly the infinitesimal version of subharmonicity on every complex line. The following criterion is therefore the main computational test for smooth plurisubharmonic functions. [quotetheorem:3403] [citeproof:3403] This criterion transforms the definition — a condition over all complex lines — into a single matrix inequality at each point. In the language of differential forms, the Levi form condition reads $i\partial\bar\partial u \geq 0$ as a $(1,1)$-form, the positive $(1,1)$-current condition. This is the starting point for the $L^2$ theory of $\bar\partial$ developed in Chapter 6, where one works directly with currents rather than smooth functions. A function $u \in C^2(\Omega)$ is called **strictly plurisubharmonic** if $\mathcal{L}_u(z)$ is strictly positive definite at every point — all eigenvalues are strictly positive. Strict psh functions give rise to strongly pseudoconvex domains (Chapter 5) and play a key role in $L^2$ estimates for the $\bar\partial$ equation. The criterion requires $u \in C^2$ in an essential way. For non-smooth psh functions — such as $\log|f|$ at the zero set of a [holomorphic function](/page/Holomorphic%20Function) $f$, where the value is $-\infty$ — the Levi form is not defined classically. One extends the criterion using distribution theory: a locally integrable $u$ is psh if and only if $i\partial\bar\partial u \geq 0$ as a current. For the computations in this chapter, smoothness holds for all examples except the maximum function, which we handle directly from the definition. ## Stability Properties Is the class $\operatorname{PSH}(\Omega)$ closed under the operations one needs in practice — taking maxima, passing to limits, composing with holomorphic maps, applying convex functions? Without such stability, the class would be fragile and difficult to use in constructions. Fortunately, the answer is yes for a natural set of operations. [quotetheorem:3404] [citeproof:3404] These stability properties form a practical toolbox. Property (4) with $\chi(t) = e^{pt}$ ($p > 0$) gives: if $u$ is psh then $e^{pu}$ is psh. With $\chi(t) = \lambda t$ for $\lambda \geq 0$, non-negative scalar multiples of psh functions are psh. For two psh functions $u_1, u_2$, their sum $u_1 + u_2$ is psh (use the Levi form criterion for smooth functions, or directly from the definition for general ones). Property (3) is the most structurally important: it guarantees that plurisubharmonicity is an intrinsic notion, unchanged by holomorphic reparametrisations. This is why psh functions can be defined on complex manifolds without reference to coordinates. The hypothesis of decreasing (not increasing) limits in property (2) is necessary. An increasing pointwise limit of psh functions is in general only lower semicontinuous, not upper semicontinuous — upper semicontinuity can fail at the limit. One can rescue property (2) for increasing sequences by taking the upper semicontinuous regularisation $u^*(z) = \limsup_{w \to z} (\lim_j u_j(w))$, which is again psh if the limit is locally bounded above. ## PSH Functions from Holomorphic Functions The abstract definition of plurisubharmonicity becomes useful only once we can produce many examples. Holomorphic functions provide the basic source: their absolute values satisfy the one-variable sub-mean-value inequality on every complex line, and taking a logarithm turns zeros into the controlled value $-\infty$. This connection lets holomorphic data generate psh weights, hulls, and exhaustion functions throughout the rest of the theory. [quotetheorem:3405] [citeproof:3405] This theorem is fundamental. It immediately gives: for any holomorphic $f$ and $p > 0$, the function $|f|^p = e^{p\log|f|}$ is psh (applying stability property (4) with $\chi(t) = e^{pt}$). More generally, for any finite collection $f_1, \ldots, f_k \in \mathcal{O}(\Omega)$, the functions \begin{align*} \log\max(|f_1|, \ldots, |f_k|) &= \max(\log|f_1|, \ldots, \log|f_k|),\\ \log\left(|f_1|^2 + \cdots + |f_k|^2\right)^{1/2} &= \frac{1}{2}\log(|f_1|^2 + \cdots + |f_k|^2) \end{align*} are psh on $\Omega$ (the first by the maximum stability property; the second requires a separate argument since it is not a maximum, but follows from the Levi form criterion as illustrated in the example below). The connection to Chapter 3 is direct: the holomorphic hull $\hat{K}_\Omega = \{z \in \Omega : |f(z)| \leq \sup_K |f| \text{ for all } f \in \mathcal{O}(\Omega)\}$ can be expressed using the psh functions $\log|f|$ — the hull is the sublevel set of the family $\{\log|f| - \sup_K \log|f| : f \in \mathcal{O}(\Omega)\}$. This connection is what links the holomorphic convexity of Chapter 3 to the psh theory developed here. ## The Boundary Blowup Function A psh function of fundamental importance in domain theory is provided by the negative log-distance to the boundary. This function will reappear in Chapter 5 as one of the defining properties of pseudoconvexity, and in Chapter 8 as a key ingredient in the proof that pseudoconvexity implies the domain-of-holomorphy property. [definition: Log-Distance Function] For a domain $\Omega \subsetneq \mathbb{C}^n$, the **log-distance function** is \begin{align*} \delta_\Omega: \Omega \to \mathbb{R}, \qquad \delta_\Omega(z) = -\log\, d(z, \partial\Omega), \end{align*} where $d(z, \partial\Omega) = \inf_{w \in \partial\Omega} |z - w|$ is the Euclidean distance from $z$ to $\partial\Omega$. [/definition] The function $\delta_\Omega$ is continuous on $\Omega$ and tends to $+\infty$ as $z \to \partial\Omega$, making it a natural boundary-control function. On bounded domains this boundary blowup is enough to give compact sublevel sets, while on unbounded domains one may also need a term controlling escape to infinity. The key question is whether this boundary-control function is merely topological, or whether it detects the analytic property of being a domain of holomorphy. The following result gives the first half of that bridge by showing that domains of holomorphy force the log-distance function to be psh. [quotetheorem:3406] [citeproof:3406] The conceptual point is that holomorphic convexity controls how compact sets sit relative to the boundary, and that control is strong enough to be detected by a psh boundary blowup function. The converse — that $\delta_\Omega$ being psh implies $\Omega$ is a domain of holomorphy — is the hard direction, proved in Chapter 8 via the $\bar\partial$-equation. [remark: Pseudoconvexity and the Log-Distance] In Chapter 5, a domain $\Omega$ is defined to be **pseudoconvex** if it admits a continuous psh exhaustion function $\phi: \Omega \to \mathbb{R}$ with $\{\phi < c\} \Subset \Omega$ for every $c \in \mathbb{R}$. The function $\delta_\Omega$ blows up to $+\infty$ at the boundary; when combined with a harmless growth term on unbounded domains, it supplies the boundary-control part of a psh exhaustion. The theorem above is therefore the first step toward the implication: every domain of holomorphy is pseudoconvex. Chapter 8 proves the converse. Together, these results give the equivalence between domains of holomorphy, holomorphic convexity, and pseudoconvexity. [/remark] ## Examples and Non-Examples We now work through the canonical examples, verifying psh directly. [example: The Squared Norm] Let $u(z) = |z|^2 = \sum_{j=1}^n |z_j|^2$ on $\mathbb{C}^n$. Since $|z_j|^2 = z_j\bar{z}_j$, the mixed partial derivatives are \begin{align*} \frac{\partial^2 u}{\partial z_\ell \partial \bar{z}_k} = \frac{\partial^2}{\partial z_\ell \partial \bar{z}_k}\sum_j z_j\bar{z}_j = \delta_{\ell k}. \end{align*} So $\mathcal{L}_u(z) = I$ (the $n \times n$ identity matrix) at every point $z \in \mathbb{C}^n$. The identity is strictly positive definite, so $u = |z|^2$ is **strictly plurisubharmonic** on all of $\mathbb{C}^n$. This is the model example of a strictly psh function. In Chapter 5, the ball $\mathbb{B}^n = \{|z| < 1\}$ will be seen to be strongly pseudoconvex precisely because the function $-\log(1 - |z|^2)$ is strictly psh on $\mathbb{B}^n$, and this in turn follows (via chain rule calculations) from the strict psh nature of $|z|^2$. [/example] [example: The Logarithm of the Squared Norm] Let $u(z_1, z_2) = \log(|z_1|^2 + |z_2|^2)$ on $\mathbb{C}^2 \setminus \{0\}$. Set $\rho = |z_1|^2 + |z_2|^2$. We compute the Levi form. Using $\partial \rho / \partial \bar{z}_k = z_k$ and the quotient rule: \begin{align*} \frac{\partial^2 \log \rho}{\partial z_j \partial \bar{z}_k} = \frac{\delta_{jk}}{\rho} - \frac{z_j \overline{z_k}}{\rho^2}. \end{align*} In matrix form, $\mathcal{L}_u(z) = \rho^{-1}(I - \rho^{-1} z \otimes \bar{z})$ where $(z \otimes \bar{z})_{jk} = z_j\overline{z_k}$. For any $\xi \in \mathbb{C}^2$, \begin{align*} \sum_{j,k} \frac{\partial^2 u}{\partial z_j \partial \bar{z}_k}\, \xi_j \overline{\xi_k} = \frac{|\xi|^2}{\rho} - \frac{|z \cdot \bar{\xi}|^2}{\rho^2}, \end{align*} where $z \cdot \bar{\xi} = \sum_j z_j\overline{\xi_j}$. By the Cauchy–Schwarz inequality, $|z \cdot \bar\xi|^2 \leq |z|^2|\xi|^2 = \rho|\xi|^2$, so \begin{align*} \frac{|\xi|^2}{\rho} - \frac{|z \cdot \bar\xi|^2}{\rho^2} \geq \frac{|\xi|^2}{\rho} - \frac{\rho|\xi|^2}{\rho^2} = 0. \end{align*} Therefore $u$ is psh on $\mathbb{C}^2 \setminus \{0\}$. Equality holds when $\xi = \lambda z$ (i.e., $\xi$ is a complex multiple of $z$), so the Levi form is not strictly positive: $u$ is psh but not strictly psh. The null direction at each $z$ is $\mathbb{C} \cdot z$ itself. This function is the standard psh function on $\mathbb{C}^n \setminus \{0\}$ and plays a key role in Chapter 9 on Reinhardt domains: in the logarithmic image, level sets of $\log|z|^2 = 2\log|z|$ become affine hyperplanes. [/example] [example: The Maximum of Logarithms] Let $u(z_1, z_2) = \max(\log|z_1|, \log|z_2|)$ on $(\mathbb{C}^*)^2 = \{z_1 z_2 \neq 0\}$. This function is not smooth (it is not differentiable where $|z_1| = |z_2|$), so the Levi form criterion does not directly apply. Instead, we use the definition directly. For fixed $z = (z_1, z_2) \in (\mathbb{C}^*)^2$ and $w = (w_1, w_2) \in \mathbb{C}^2$, the restriction of $u$ to the complex line through $z$ in direction $w$ is \begin{align*} \zeta \mapsto \max\bigl(\log|z_1 + \zeta w_1|, \log|z_2 + \zeta w_2|\bigr). \end{align*} Each function $\zeta \mapsto \log|z_j + \zeta w_j|$ is the log-modulus of a [holomorphic function](/page/Holomorphic%20Function) of $\zeta$, hence subharmonic. The pointwise maximum of two subharmonic functions is subharmonic. Therefore the restriction is subharmonic for every $(z, w)$, and $u \in \operatorname{PSH}((\mathbb{C}^*)^2)$. The zero set $\{z_j = 0\}$ can be included with value $-\infty$, and the same argument applies. Note that $\max(\log|z_1|, \log|z_2|) < 0$ if and only if $|z_1| < 1$ and $|z_2| < 1$, so the sublevel set $\{u < 0\} = \mathbb{D}^2$ is the bidisc. In Chapter 9, this function appears as the logarithmic indicator of the polydisc, connecting Reinhardt domain theory to the psh class. [/example] [example: Subharmonic but Not Plurisubharmonic] We exhibit a function that is subharmonic on $\mathbb{C}^2 \cong \mathbb{R}^4$ but not plurisubharmonic. Define \begin{align*} u(z_1, z_2) = \operatorname{Re}(z_1 \bar{z}_2) = x_1 x_2 + y_1 y_2, \end{align*} where $z_j = x_j + iy_j$. The function $u = x_1 x_2 + y_1 y_2$ is a polynomial of degree 2 in $\mathbb{R}^4$; its real Hessian is indefinite, but the Laplacian $\Delta_{\mathbb{R}^4} u = 0$ (all second-order terms vanish: $\partial^2 u/\partial x_1^2 = \partial^2 u / \partial y_1^2 = 0$, and similarly for $x_2, y_2$). So $u$ is harmonic on $\mathbb{R}^4$, and in particular subharmonic. We compute the Levi form. Since $u = \frac{1}{2}(z_1\bar{z}_2 + \bar{z}_1 z_2)$, we differentiate using the Wirtinger operators. We have $\partial u/\partial \bar{z}_1 = z_2/2$ and $\partial u/\partial \bar{z}_2 = z_1/2$. Then: \begin{align*} \frac{\partial^2 u}{\partial z_1 \partial \bar{z}_1} = 0, \quad \frac{\partial^2 u}{\partial z_2 \partial \bar{z}_1} = \frac{1}{2}, \quad \frac{\partial^2 u}{\partial z_1 \partial \bar{z}_2} = \frac{1}{2}, \quad \frac{\partial^2 u}{\partial z_2 \partial \bar{z}_2} = 0. \end{align*} So the Levi form is the constant matrix \begin{align*} \mathcal{L}_u = \begin{pmatrix} 0 & \frac{1}{2} \\ \frac{1}{2} & 0 \end{pmatrix}, \end{align*} with eigenvalues $\pm 1/2$. Since one eigenvalue is $-1/2 < 0$, the matrix is not positive semidefinite. Taking $\xi = (1, -1) \in \mathbb{C}^2$: \begin{align*} \sum_{j,k} \mathcal{L}_u{}_{jk}\, \xi_j \overline{\xi_k} = 0 - \frac{1}{2} - \frac{1}{2} + 0 = -1 < 0. \end{align*} Therefore $u$ is **not** plurisubharmonic, despite being harmonic (and hence subharmonic) on $\mathbb{R}^4$. Concretely: the complex line $\{(0, \zeta) : \zeta \in \mathbb{C}\}$ (i.e., $z = 0$, $w = (0,1)$) gives the restriction $\zeta \mapsto u(0, \zeta) = \operatorname{Re}(0 \cdot \bar\zeta) = 0$, which is harmonic. But the complex line $\{(\zeta, -\zeta) : \zeta \in \mathbb{C}\}$ (i.e., $z = 0$, $w = (1,-1)$) gives $\zeta \mapsto u(\zeta, -\zeta) = \operatorname{Re}(\zeta \cdot \overline{(-\zeta)}) = \operatorname{Re}(-|\zeta|^2) = -|\zeta|^2$, which is strictly superharmonic ($\Delta(-|\zeta|^2) = -4 < 0$). This directly violates the definition of psh. [/example] The non-example makes precise why the restriction to complex lines is essential: the real Laplacian on $\mathbb{R}^{2n}$ is insensitive to the complex structure, mixing holomorphic and anti-holomorphic derivatives, while the Levi form isolates only the holomorphic contribution. Any class of functions compatible with holomorphic geometry must impose conditions on the complex Hessian, not just on the real Laplacian. ## Looking Ahead to Pseudoconvexity The psh functions developed in this chapter are the raw material for Chapter 5. A domain $\Omega \subset \mathbb{C}^n$ will be called pseudoconvex if it carries a psh exhaustion function — a psh function that is proper, tends to $+\infty$ at the boundary, and also controls escape to infinity when the domain is unbounded. The function $\delta_\Omega = -\log d(\cdot, \partial\Omega)$ is the canonical boundary-control candidate, and the theorem above shows it is psh for domains of holomorphy. The Levi form criterion will reappear in the boundary form of pseudoconvexity: for a domain with smooth boundary defined by $\rho < 0$, the boundary is Levi pseudoconvex at a point $p \in \partial\Omega$ if the Levi form of $\rho$ restricted to the complex tangent space $\ker(\partial\rho(p)) \cap T_p^{1,0}(\partial\Omega)$ is positive semidefinite. This is the direct boundary analogue of the interior condition $\mathcal{L}_u \succeq 0$. The stability under holomorphic pullback (property (3)) will justify defining pseudoconvexity intrinsically on complex manifolds in later courses: because psh functions pull back under holomorphic maps, the psh condition and hence pseudoconvexity are biholomorphic invariants. The convex composition property (property (4)) will allow one to modify psh exhaustions — for instance, replacing $\phi$ by $e^\phi$ or $\chi \circ \phi$ for appropriate $\chi$ — to achieve strict positivity of the Levi form, which is needed for the $L^2$ estimates in Chapter 6. --- Pseudoconvexity is the translation of plurisubharmonicity into intrinsic domain geometry: a domain is pseudoconvex when its sublevel sets with respect to an exhaustion function are plurisubharmonic. This biholomorphic invariant becomes the cornerstone for solving the Levi problem, but only through the analytic machinery of the $\bar\partial$-equation can pseudoconvexity be shown to force a domain to be a domain of holomorphy. # 5. Pseudoconvexity Pseudoconvexity is the central geometric condition linking boundary curvature to [analytic continuation](/page/Analytic%20Continuation). Chapter 4 equipped us with plurisubharmonic functions and showed that $-\log d(\cdot, \partial\Omega)$ is psh on every domain of holomorphy; this chapter asks what it means for a domain to be shaped so that such a psh witness exists. The answer is pseudoconvexity, which can be stated either analytically (via psh exhaustion functions) or geometrically (via the Levi form at smooth boundary points). The central question — whether the two formulations agree with each other and with the domain-of-holomorphy condition — is the Levi problem. Its solution, deferred to Chapter 8, requires the $\bar\partial$-machinery of Chapter 6, but this chapter lays the precise definitions and establishes the easy direction of the equivalence. ## Pseudoconvexity via Exhaustion Functions The analytic definition of pseudoconvexity requires the existence of a plurisubharmonic function that becomes large near the boundary of the domain, controlling the geometry from the inside. Such a function is called an exhaustion function. To motivate the condition, recall from Chapter 4 that a domain of holomorphy $\Omega$ has the property that $-\log d(\cdot, \partial\Omega)$ is plurisubharmonic. This function blows up to $+\infty$ as $z \to \partial\Omega$, so on bounded domains its sublevel sets stay compactly inside $\Omega$. On unbounded domains, one supplements boundary control with growth at infinity. If pseudoconvexity is to be equivalent to being a domain of holomorphy, then the existence of some psh function with this full exhaustion property — not necessarily the specific log-distance function alone — should be the right condition. [definition: Pseudoconvex Domain] A domain $\Omega \subset \mathbb{C}^n$ is **pseudoconvex** if there exists a continuous plurisubharmonic function $\phi: \Omega \to \mathbb{R}$ such that for every $c \in \mathbb{R}$, the sublevel set \begin{align*} \Omega_c = \{z \in \Omega : \phi(z) < c\} \end{align*} is relatively compact in $\Omega$, i.e., $\overline{\Omega_c} \subset \Omega$ is compact. Such a function $\phi$ is called a **psh exhaustion function** for $\Omega$. [/definition] The compactness condition $\Omega_c \Subset \Omega$ means that $\phi(z) \to +\infty$ as $z \to \partial\Omega$ (or as $|z| \to \infty$ if $\Omega$ is unbounded). The sublevel sets serve as relatively compact approximations of $\Omega$ from within: $\Omega = \bigcup_{c} \Omega_c$ and each closure $\overline{\Omega_c}$ is compact in $\Omega$. This makes $\phi$ a natural tool for producing compact exhaustions of $\Omega$, which arise throughout the theory whenever one needs to apply arguments valid on compact sets and then pass to the limit. The requirement that $\phi$ be plurisubharmonic — rather than merely continuous, or subharmonic in the real sense — is the critical restriction. A continuous real-valued exhaustion function exists for any domain (e.g., $-\log d(\cdot, \partial\Omega)$), but without the psh condition it carries no complex-analytic information. [remark: Continuity vs. Smoothness] The definition asks for a continuous psh exhaustion. In practice, one can often arrange the exhaustion to be smooth, and the smooth case connects directly to the Levi pseudoconvexity formulation introduced in the next section. For the purposes of Definition, continuity suffices and allows the non-smooth examples that arise naturally in the theory. [/remark] ## The Levi Pseudoconvexity Condition The exhaustion definition of pseudoconvexity is analytically natural but difficult to apply directly: given a domain, constructing an explicit global psh exhaustion function requires global information about the domain, and verifying plurisubharmonicity globally is not a routine computation. What one wants instead is a pointwise condition at the boundary that can be checked locally — a condition that reduces the global question to a computation at each boundary point. Without such a reduction, pseudoconvexity is not a condition one can practically verify for specific examples, and the connection to boundary geometry remains opaque. The Levi condition provides exactly this: a criterion phrased entirely in terms of the second-order complex geometry of $\partial\Omega$ at each point. Let $\Omega \subset \mathbb{C}^n$ be a bounded domain with $C^2$ boundary. Near each boundary point $p \in \partial\Omega$, the boundary can be described as the zero set of a smooth function: $\partial\Omega = \{\rho = 0\}$ where $\rho: U \to \mathbb{R}$ is $C^2$ on a neighbourhood $U$ of $p$ with $\nabla \rho \neq 0$ on $\partial\Omega$ and $\Omega \cap U = \{z \in U : \rho(z) < 0\}$. Such $\rho$ is called a **defining function** for $\Omega$ near $p$. At each boundary point $p \in \partial\Omega$, the tangent space $T_p(\partial\Omega)$ (as a real $(2n-1)$-dimensional hyperplane) contains a maximal complex subspace, the **complex tangent space**. Concretely, a holomorphic tangent vector at $p$ is a vector $w \in \mathbb{C}^n$ tangent to $\partial\Omega$ in the complex-linear sense: \begin{align*} T_p^{1,0}(\partial\Omega) = \left\{ w \in \mathbb{C}^n : \sum_{j=1}^n \frac{\partial \rho}{\partial z_j}(p)\, w_j = 0 \right\}. \end{align*} This is a complex $(n-1)$-dimensional subspace of $\mathbb{C}^n$. It is the space of directions in which one can move along $\partial\Omega$ while keeping the movement complex-linear. The **Levi form** of the defining function $\rho$ at a boundary point $p$ is the Hermitian form on $T_p^{1,0}(\partial\Omega)$ defined by restricting the complex Hessian of $\rho$: \begin{align*} \mathcal{L}_\rho(p; w) = \sum_{j,k=1}^n \frac{\partial^2 \rho}{\partial z_j \partial \bar{z}_k}(p)\, w_j \overline{w_k}, \quad w \in T_p^{1,0}(\partial\Omega). \end{align*} This is the same matrix studied in Chapter 4 as the Levi form of a function, now evaluated specifically at boundary points and restricted to the complex tangent directions. To see why positive semidefiniteness matters, consider what happens when the Levi form is indefinite. Take the domain $\Omega = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1|^2 - |z_2|^2 < 1\}$ with defining function $\rho(z) = |z_1|^2 - |z_2|^2 - 1$. At the boundary point $p = (1, 0)$, the complex Hessian of $\rho$ is the diagonal matrix $\operatorname{diag}(1, -1)$. A complex tangent vector $w \in T_p^{1,0}(\partial\Omega)$ satisfies $\bar z_1 w_1 = 0$, i.e., $w_1 = 0$, so $w = (0, w_2)$ with $w_2$ free. Then $\mathcal{L}_\rho(p; w) = -|w_2|^2 < 0$ for $w_2 \neq 0$. The Levi form is negative on a nonzero tangent vector, and indeed $\Omega$ is not pseudoconvex: it is biholomorphic to the complement of a ball, which is not a domain of holomorphy. Indefiniteness of the Levi form signals that one can approach the boundary from inside along complex curves in ways that violate the exhaustion property. [definition: Levi Pseudoconvex Domain] A domain $\Omega \subset \mathbb{C}^n$ with $C^2$ boundary is **Levi pseudoconvex** if, for every boundary point $p \in \partial\Omega$ and every vector $w \in T_p^{1,0}(\partial\Omega)$, \begin{align*} \mathcal{L}_\rho(p; w) = \sum_{j,k=1}^n \frac{\partial^2 \rho}{\partial z_j \partial \bar{z}_k}(p)\, w_j \overline{w_k} \geq 0. \end{align*} If the inequality is strict for all $w \neq 0$, the domain is called **strictly Levi pseudoconvex**. [/definition] The Levi pseudoconvexity condition says that the complex Hessian of the defining function, restricted to the complex tangent space, is positive semidefinite at every boundary point. The restriction to $T_p^{1,0}(\partial\Omega)$ is essential: the Levi form of $\rho$ is positive semidefinite on all of $\mathbb{C}^n$ (i.e., $\rho$ is psh) would be a much stronger condition, not satisfied in general. The condition is independent of the choice of defining function $\rho$ in the following sense: if $\tilde\rho = e^\psi \rho$ is another defining function for the same domain ($\psi$ smooth), then the Levi form of $\tilde\rho$ restricted to $T_p^{1,0}(\partial\Omega)$ is $e^{\psi(p)}$ times the Levi form of $\rho$. Since $e^{\psi(p)} > 0$, the sign — and hence positive semidefiniteness — is preserved. The condition is therefore a property of the domain, not of the particular defining function. ## Equivalence for Smooth Bounded Domains We now have two definitions of pseudoconvexity: the exhaustion definition (analytic, global, intrinsic to the domain) and the Levi condition (geometric, local, defined only at smooth boundary points). These are logically independent — one lives on the interior, the other on the boundary. If they were not equivalent, the theory would fracture into two separate notions that might diverge for concrete domains, and one would have to specify which kind of pseudoconvexity is meant in every theorem. Establishing their equivalence for smooth bounded domains is therefore not merely a convenience; it is what makes "pseudoconvexity" a single coherent concept. [quotetheorem:3407] [citeproof:3407] The equivalence justifies using either characterisation depending on which is more convenient for the problem at hand. Geometric arguments typically use the Levi form, while analytic constructions (such as the $L^2$ approach to the $\bar\partial$-equation) use the exhaustion. The $C^2$ boundary hypothesis is not merely a technical convenience — it is essential for the Levi condition to be defined at all. The Levi form involves second-order derivatives of the defining function $\rho$, so at a boundary point where $\rho$ fails to be twice differentiable, the form is not defined. For domains with corners (such as the polydisc $\mathbb{D}^n$) or with $C^1$ but not $C^2$ boundary, the Levi condition cannot be checked pointwise at the singular boundary points. The exhaustion definition remains valid for such domains, and $\mathbb{D}^n$ is indeed pseudoconvex (one exhaustion is $\phi(z) = \max_j (-\log(1 - |z_j|^2))$, a maximum of psh functions and hence psh), but one cannot verify this by checking the Levi form at the corner points $\{|z_1| = |z_2| = \dots = 1\}$. For non-smooth domains, the equivalence theorem simply does not apply — the Levi formulation is unavailable. To illustrate both formulations on the same domain, consider the unit ball $\mathbb{B}^n$. From the exhaustion side, $\phi = -\log(1 - |z|^2)$ is psh (verified by computing the Levi form of $\phi$ in the next section) with compact sublevel sets $\{|z|^2 < 1 - e^{-c}\}$, so $\mathbb{B}^n$ is pseudoconvex in the exhaustion sense. From the Levi side, the defining function is $\rho = |z|^2 - 1$, the complex Hessian of $\rho$ is the identity matrix $(\partial^2 \rho / \partial z_j \partial \bar z_k) = \delta_{jk}$, and for any complex tangent vector $w$ satisfying $\sum_j \bar z_j w_j = 0$, the Levi form evaluates to $|w|^2 \geq 0$. Both formulations confirm pseudoconvexity and in fact strict Levi pseudoconvexity, consistent with the equivalence theorem. ## The Levi Problem Having two equivalent formulations of pseudoconvexity, the natural question is how this condition relates to the domain-of-holomorphy condition introduced in Chapter 3. The connection runs through the psh function $-\log d(\cdot, \partial\Omega)$: by the result from Chapter 4, this function is psh on every domain of holomorphy (a consequence of the [Cartan–Thullen theorem](/theorems/3385) from Chapter 3, via the boundary [distance characterisation of the holomorphic hull](/theorems/3402)). Since $-\log d(z, \partial\Omega) \to +\infty$ as $z \to \partial\Omega$, it supplies the needed boundary control; on unbounded domains, an additional psh growth term controls infinity. This gives the implication that every domain of holomorphy is pseudoconvex. The reverse direction is the hard part. [quotetheorem:3386] [citeproof:3386] The reverse question is harder than it appears. The proof above passes through a single specific psh function — the log-distance function — and uses the [Cartan–Thullen theorem](/theorems/3385) as a black box. Reversing the argument would require: given any continuous psh exhaustion $\phi$ (not necessarily the log-distance), construct a [holomorphic function](/page/Holomorphic%20Function) on $\Omega$ that cannot be continued. But the exhaustion $\phi$ is not holomorphic, and there is no obvious way to produce holomorphic functions from a real-valued psh function alone. The obstruction is that pseudoconvexity is a condition on a function class (psh functions), while being a domain of holomorphy is a condition on holomorphic functions, and bridging the two requires a deeper mechanism. The $\bar\partial$-equation provides that mechanism: on a pseudoconvex domain one can solve $\bar\partial u = f$ with optimal $L^2$ estimates (Hörmander's theorem), and from such solutions one constructs the necessary holomorphic functions. This is why the proof of the Levi problem is substantially harder than the forward direction. The reverse direction deserves its own name because it asks for a bridge from a real-variable convexity condition to the existence of genuinely holomorphic obstruction functions. This formulation becomes the organizing problem for the later analytic machinery: if pseudoconvexity is the right intrinsic replacement for ordinary convexity, it should exactly characterize domains of holomorphy. [definition: The Levi Problem] The **Levi problem** asks: is every pseudoconvex domain a domain of holomorphy? [/definition] Equivalently, given a domain $\Omega$ admitting a continuous psh exhaustion function, does there exist a [holomorphic function](/page/Holomorphic%20Function) $f \in \mathcal{O}(\Omega)$ that cannot be continued beyond any boundary point? The question was posed by Levi in 1910, and remained open for more than forty years. Its resolution by Oka (1942), Bremermann, and Norguet (1954) is the main theorem of Chapter 8. The answer is yes: pseudoconvexity and the domain-of-holomorphy condition are equivalent. The proof requires either the theory of the $\bar\partial$-equation with $L^2$ estimates (Hörmander's approach, developed in later courses) or Oka's original sheaf-theoretic method. For now, the three characterisations of domains of holomorphy — holomorphic convexity (Cartan–Thullen), psh exhaustion existence, and Levi pseudoconvexity for smooth boundaries — are all equivalent, and this chapter establishes the third definition precisely so that its equivalence with the others can be stated cleanly in Chapter 8. ## Convex Domains Are Pseudoconvex At this point the definitions are abstract: pseudoconvexity requires a psh exhaustion function to exist, but there is no general recipe for constructing one. For which domains can one write down such a function explicitly? The definition gives no guidance on its own — it merely asserts existence. Before proceeding to more subtle examples, one needs a class of domains where pseudoconvexity can be established from first principles, without ad hoc constructions. Convex domains provide that class: their boundary distance has the right psh behaviour, and on unbounded convex domains one adds a growth term to obtain a genuine exhaustion. [quotetheorem:3408] [citeproof:3408] This result places ordinary convexity inside the broader complex notion of pseudoconvexity. The inclusion is strict: plurisubharmonic functions need not be convex, and pseudoconvex domains need not be convex. Convex domains should therefore be viewed as a reliable testing ground for the theory rather than as the full geometric picture. ## The Siegel Domain Having established that [convex domains are pseudoconvex](/theorems/3408), it is worth examining a specific example that connects pseudoconvexity to the study of automorphisms and biholomorphic equivalences. The Siegel upper half-space is an unbounded convex domain that is biholomorphic to the unit ball, providing a useful alternative model for the ball with a parabolic boundary structure. [example: The Siegel Domain is Pseudoconvex] Define the **Siegel domain** by \begin{align*} \mathcal{S} = \{ (z_1, z_2) \in \mathbb{C}^2 : \operatorname{Im}(z_2) > |z_1|^2 \}. \end{align*} This is an unbounded domain in $\mathbb{C}^2$. The convexity can be checked directly. If $p,q\in\mathcal{S}$ and $0\leq t\leq 1$, then \begin{align*} \operatorname{Im}(t p_2+(1-t)q_2) &=t\,\operatorname{Im}p_2+(1-t)\operatorname{Im}q_2\\ &>t|p_1|^2+(1-t)|q_1|^2\\ &\geq |t p_1+(1-t)q_1|^2, \end{align*} where the last inequality is convexity of the squared norm. Hence $tp+(1-t)q\in\mathcal{S}$, so $\mathcal{S}$ is convex. By the result of the previous section, [convex domains are pseudoconvex](/theorems/3408). To verify pseudoconvexity, define the function \begin{align*} \phi(z_1, z_2) = -\log(\operatorname{Im}(z_2) - |z_1|^2). \end{align*} We claim $\phi$ is psh on $\mathcal{S}$. Note that $\phi$ blows up to $+\infty$ as $(z_1, z_2) \to \partial\mathcal{S}$ (since $\operatorname{Im}(z_2) - |z_1|^2 \to 0^+$ there), but the sublevel sets $\{\phi < c\} = \{\operatorname{Im}(z_2) - |z_1|^2 > e^{-c}\}$ are not compact in $\mathcal{S}$ because $\mathcal{S}$ is unbounded: taking $z_1 = t$ real and $z_2 = (t^2 + 1 + e^{-c}/2)i$ shows that points with $|z_1| \to \infty$ can still satisfy $\operatorname{Im}(z_2) - |z_1|^2 > e^{-c}$. The function $\phi$ alone is therefore not a psh exhaustion. To obtain an actual exhaustion one can use $\Phi(z) = \phi(z) + |z|^2$, which is psh (sum of psh functions) and has compact sublevel sets since $|z|^2$ forces boundedness. Alternatively, and more cleanly, pseudoconvexity of $\mathcal{S}$ follows from convexity (established above) via the general theorem. The computation below, however, is still useful: it gives an explicit psh function on $\mathcal{S}$ whose Levi form is strictly positive, providing a direct check that $\mathcal{S}$ is strictly pseudoconvex. It remains to verify that $\phi$ is psh. Write $h(z) = \operatorname{Im}(z_2) - |z_1|^2 = \frac{1}{2i}(z_2 - \bar z_2) - |z_1|^2$. Then $\phi = -\log h$. We compute the Levi form. Noting that $\partial h / \partial \bar z_1 = -z_1$, $\partial h / \partial \bar z_2 = -1/(2i) = i/2$, and differentiating again: \begin{align*} \frac{\partial^2 h}{\partial z_j \partial \bar z_k}: \quad \frac{\partial^2 h}{\partial z_1 \partial \bar z_1} = -1, \quad \frac{\partial^2 h}{\partial z_j \partial \bar z_k} = 0 \text{ for } (j,k) \neq (1,1). \end{align*} For $\phi = -\log h$, the chain rule gives \begin{align*} \frac{\partial^2 \phi}{\partial z_j \partial \bar z_k} = -\frac{1}{h}\frac{\partial^2 h}{\partial z_j \partial \bar z_k} + \frac{1}{h^2}\frac{\partial h}{\partial z_j}\overline{\frac{\partial h}{\partial z_k}}. \end{align*} The Levi form applied to $\xi = (\xi_1, \xi_2) \in \mathbb{C}^2$ therefore gives \begin{align*} \mathcal{L}_\phi(\xi, \xi) = \frac{|\xi_1|^2}{h} + \frac{1}{h^2}\left| -z_1\xi_1 + \frac{i}{2}\xi_2 \right|^2 \geq 0, \end{align*} since both terms are nonneg­ative. Thus $\phi$ is psh, confirming that $\mathcal{S}$ is pseudoconvex. [/example] The Siegel domain is biholomorphic to the unit ball $\mathbb{B}^2$ via a Cayley transform after the standard normalisation of the half-space coordinate. This fact will be important when studying automorphism groups in later work: the transform sends the parabolic boundary $\{\operatorname{Im}(z_2) = |z_1|^2\}$ to the sphere $\partial\mathbb{B}^2$. Since biholomorphisms preserve pseudoconvexity (a psh exhaustion on one domain pulls back to a psh exhaustion on the other), the pseudoconvexity of $\mathcal{S}$ is consistent with the pseudoconvexity of $\mathbb{B}^2$. This also shows that strict Levi pseudoconvexity is biholomorphically invariant — both $\partial\mathcal{S}$ and $\partial\mathbb{B}^2$ are strictly Levi pseudoconvex hypersurfaces, as one can verify directly. We now verify the pseudoconvexity of $\mathbb{B}^n$ explicitly and then contrast it with the Hartogs figure, which illustrates how pseudoconvexity can fail. ## The Unit Ball and the Hartogs Figure The theorems so far tell us that [convex domains are pseudoconvex](/theorems/3408) and that every domain of holomorphy is pseudoconvex. But the unit ball $\mathbb{B}^n$ is convex, so its pseudoconvexity follows from the preceding theorem — yet convexity alone does not produce the sharpest understanding of its boundary geometry. A more pressing question is whether one can verify pseudoconvexity for specific well-known domains — directly, by exhibiting an explicit exhaustion and computing that it is psh — and whether one can identify a specific domain that fails to be pseudoconvex. The unit ball and the Hartogs figure answer both questions concretely. [example: The Unit Ball is Pseudoconvex] The unit ball $\mathbb{B}^n = \{z \in \mathbb{C}^n : |z|^2 < 1\}$ is pseudoconvex. Define \begin{align*} \phi(z) = -\log(1 - |z|^2). \end{align*} As $z \to \partial\mathbb{B}^n$, we have $|z|^2 \to 1$, so $1 - |z|^2 \to 0^+$ and $\phi \to +\infty$. The sublevel set $\{\phi < c\} = \{|z|^2 < 1 - e^{-c}\}$ is the ball of radius $(1 - e^{-c})^{1/2}$, which is compactly contained in $\mathbb{B}^n$. It remains to verify that $\phi$ is psh. Write $\phi = -\log(1 - |z|^2)$. Setting $s = |z|^2 = \sum_j |z_j|^2$ and $h = 1 - s$, we compute \begin{align*} \frac{\partial^2 \phi}{\partial z_j \partial \bar z_k} = \frac{\delta_{jk}}{h} + \frac{z_j \overline{z_k}}{h^2}. \end{align*} For any $\xi \in \mathbb{C}^n$, the Levi form gives \begin{align*} \mathcal{L}_\phi(\xi, \xi) = \frac{|\xi|^2}{h} + \frac{|z \cdot \bar\xi|^2}{h^2} \geq 0, \end{align*} where $z \cdot \bar\xi = \sum_j z_j \overline{\xi_j}$. Both terms are nonneg­ative, so $\phi$ is psh. Therefore $\mathbb{B}^n$ is pseudoconvex with exhaustion $-\log(1 - |z|^2)$. [/example] The unit ball is in fact a domain of holomorphy (this can also be verified directly by exhibiting a function holomorphic on $\mathbb{B}^n$ that does not extend across any boundary point), consistent with the Levi problem solution. The Levi pseudoconvexity of $\partial\mathbb{B}^n$ can be verified directly: the boundary $\partial\mathbb{B}^n = \{|z|^2 = 1\}$ has defining function $\rho(z) = |z|^2 - 1$, and the Levi form of $\rho$ is the identity matrix $\mathcal{L}_\rho = I$. For $w \in T_p^{1,0}(\partial\mathbb{B}^n)$, the condition $\sum_j \bar z_j w_j = 0$ must hold, and $\mathcal{L}_\rho(p; w) = |w|^2 \geq 0$. Thus $\partial\mathbb{B}^n$ is in fact strictly Levi pseudoconvex. [example: The Hartogs Figure is Not Pseudoconvex] Fix parameters $r = 1/2$ and $s = 1/4$, and define the **Hartogs figure** \begin{align*} H = \{ (z_1, z_2) \in \mathbb{D}^2 : |z_1| > 1/2 \} \cup \{ (z_1, z_2) \in \mathbb{D}^2 : |z_2| < 1/4 \}. \end{align*} We claim $H$ is not pseudoconvex, equivalently, not a domain of holomorphy. The Hartogs extension theorem from Chapter 2 says that every $f \in \mathcal{O}(H)$ extends holomorphically to the full bidisc $\mathbb{D}^2$, so no [holomorphic function](/page/Holomorphic%20Function) on $H$ can be singular at every interior boundary point of the missing region. To verify that $H \subset \mathbb{D}^2$: both pieces satisfy $|z_1| < 1$ and $|z_2| < 1$ (the first by definition of $\mathbb{D}^2$, the second explicitly). We also check that $H$ is a proper subdomain: the point $(0, 1/2)$ satisfies $|z_1| = 0 \leq 1/2$ (not in the first piece) and $|z_2| = 1/2 \geq 1/4$ (not in the second piece), so $(0, 1/2) \in \mathbb{D}^2 \setminus H$. The extension mechanism is the same Hartogs phenomenon from Chapter 2: the shell-and-cylinder geometry forces the missing core to be filled in. This should not be confused with the compact Hartogs--Bochner theorem; the Hartogs figure is a special noncompact local model inside the bidisc. This is enough for the example: $H$ cannot be a domain of holomorphy, since its holomorphic functions do not see the interior hole as a genuine boundary. [/example] The Hartogs figure is the prototypical example of a domain that fails the necessary condition for being a domain of holomorphy: every [holomorphic function](/page/Holomorphic%20Function) on $H$ extends to the strictly larger domain $\mathbb{D}^2$. The failure is caused by the special shell-and-cylinder geometry of $H$, not by a compact connected-complement removability theorem. From the pseudoconvexity perspective, the missing middle region creates a Hartogs-type indentation that cannot support a psh exhaustion compatible with holomorphic convexity. ## Summary The chapter introduces pseudoconvexity in two equivalent ways for smooth bounded domains. The **exhaustion definition** — existence of a continuous psh function with compact sublevel sets — is the analytic formulation suited to global constructions. The **Levi condition** — positive semidefiniteness of the restricted complex Hessian of the defining function — is the local geometric condition at the boundary. Their equivalence for $C^2$ bounded domains shows that pseudoconvexity can be checked pointwise at the boundary; for non-smooth domains such as the polydisc, only the exhaustion definition is available. Every domain of holomorphy is pseudoconvex (proved via the log-distance exhaustion), and the Levi problem — whose resolution requires the $\bar\partial$-equation — asks whether the converse holds; the affirmative answer is the main theorem of Chapter 8. Convex domains are always pseudoconvex, and the unit ball $\mathbb{B}^n$ (exhausted by $-\log(1 - |z|^2)$, strictly Levi pseudoconvex) and the Siegel domain $\mathcal{S} = \{\operatorname{Im}(z_2) > |z_1|^2\}$ (convex, pseudoconvex via the convexity theorem, biholomorphic to $\mathbb{B}^2$) are the key worked examples. The Hartogs figure $H = \{|z_1| > 1/2\} \cup \{|z_2| < 1/4\}$ in $\mathbb{D}^2$ illustrates that pseudoconvexity can fail for explicit domains: every [holomorphic function](/page/Holomorphic%20Function) on $H$ extends to $\mathbb{D}^2$ by the Hartogs extension theorem, so $H$ is not a domain of holomorphy. --- The $\bar\partial$-equation provides the analytic tool to exploit pseudoconvexity: when a domain is pseudoconvex, the $\bar\partial$-equation becomes solvable, and this solvability feeds directly into the Cousin problems. These classical problems—interpolating and factorizing holomorphic functions—are the pathway from pseudoconvexity to the domain-of-holomorphy condition and thus to the full resolution of the Levi problem. # 6. The $\bar\partial$-Equation and Dolbeault Cohomology The Dolbeault framework is the analytic backbone that connects complex geometry to the solvability of global analytic problems. The central operator is $\bar\partial$, the Cauchy–Riemann operator in several variables, and the question of when the equation $\bar\partial u = f$ can be solved governs whether domains carry enough holomorphic functions to solve the Cousin problems studied in Chapter 7. This chapter introduces the algebra of $(p,q)$-forms, proves the fundamental local solvability result (the $\bar\partial$-Poincaré lemma), defines the Dolbeault cohomology groups that measure global obstructions, and states the Stein-vanishing theorem that later interacts with the Levi problem. ## The $\bar\partial$ Operator and $(p,q)$-Forms In one complex variable, the Cauchy–Riemann condition says that a smooth function $f: U \subset \mathbb{C} \to \mathbb{C}$ is holomorphic precisely when $\partial_{\bar z} f = 0$. The problem in several variables is more subtle: the exterior algebra of $\mathbb{C}^n$ decomposes into bidegrees $(p,q)$ according to the number of holomorphic versus anti-holomorphic differentials, and one wants to detect holomorphicity at each level of this decomposition. The $\bar\partial$ operator is the part of the [exterior derivative](/theorems/1525) that raises the anti-holomorphic degree, and understanding its kernel and image in every bidegree is the analytic core of several complex variables. To define it, one must first understand what forms adapted to the complex structure look like. On $\mathbb{C}^n$, with coordinates $z_j = x_j + iy_j$, the complex cotangent space at each point is spanned by the $(1,0)$-forms $dz_j = dx_j + i\,dy_j$ and the $(0,1)$-forms $d\bar z_j = dx_j - i\,dy_j$, for $j = 1, \dots, n$. These decompose the exterior algebra according to how many factors of each type appear. [definition: $(p,q)$-Form] A **differential form of type $(p,q)$** on an [open set](/page/Open%20Set) $\Omega \subset \mathbb{C}^n$ is a smooth section of the bundle $\Lambda^{p,q} T^*\Omega$, where $\Lambda^{p,q} T^*\Omega$ is spanned at each point by wedge products \begin{align*} dz_{j_1} \wedge \cdots \wedge dz_{j_p} \wedge d\bar z_{k_1} \wedge \cdots \wedge d\bar z_{k_q} \end{align*} with $1 \leq j_1 < \cdots < j_p \leq n$ and $1 \leq k_1 < \cdots < k_q \leq n$. The space of smooth $(p,q)$-forms on $\Omega$ is denoted $\mathcal{E}^{p,q}(\Omega)$. [/definition] Every smooth $(p,q)$-form $\alpha \in \mathcal{E}^{p,q}(\Omega)$ can be written uniquely as \begin{align*} \alpha = \sum_{|J|=p,\, |K|=q} a_{JK}\, dz_J \wedge d\bar z_K, \end{align*} where $J = (j_1, \dots, j_p)$ and $K = (k_1, \dots, k_q)$ are increasing multi-indices, $dz_J = dz_{j_1} \wedge \cdots \wedge dz_{j_p}$, and $a_{JK}: \Omega \to \mathbb{C}$ are smooth functions. This decomposition is canonical: the $(p,q)$-type of a form is an intrinsic piece of data, not a choice of basis. The decomposition into $(p,q)$-types creates a new problem: the exterior derivative $d$ mixes types, so it is too coarse to isolate the Cauchy–Riemann equations on forms. We need the component of $d$ that differentiates only in the anti-holomorphic directions, because that component turns holomorphicity into a kernel condition and later turns solvability questions into cohomology. [definition: The $\bar\partial$ Operator] The **Cauchy–Riemann operator** $\bar\partial: \mathcal{E}^{p,q}(\Omega) \to \mathcal{E}^{p,q+1}(\Omega)$ is defined on a form $\alpha = \sum_{J,K} a_{JK}\, dz_J \wedge d\bar z_K$ by \begin{align*} \bar\partial \alpha = \sum_{J,K} \bar\partial a_{JK} \wedge dz_J \wedge d\bar z_K, \end{align*} where for a smooth function $f$, \begin{align*} \bar\partial f = \sum_{k=1}^n \frac{\partial f}{\partial \bar z_k}\, d\bar z_k, \end{align*} with Wirtinger derivative $\frac{\partial}{\partial \bar z_k} = \frac{1}{2}\!\left(\frac{\partial}{\partial x_k} + i\frac{\partial}{\partial y_k}\right)$ (as defined in Chapter 1). [/definition] The operator $\bar\partial$ raises the $(0,q)$ bidegree by one: it takes $(p,q)$-forms to $(p,q+1)$-forms. This is because $\bar\partial a_{JK}$ is a $(0,1)$-form, and wedging a $(0,1)$-form with a $(p,q)$-form yields a $(p,q+1)$-form. Once $\bar\partial$ is defined on forms, the next issue is whether repeated application produces a coherent complex. For cohomology to make sense, every $\bar\partial$-exact form must automatically be $\bar\partial$-closed. That requirement is exactly the algebraic identity that $\bar\partial$ squares to zero. [quotetheorem:3409] [citeproof:3409] [explanation: $\bar\partial^2 = 0$ in Context] The identity $\bar\partial^2 = 0$ is the structural compatibility condition for the $\bar\partial$-operator. It lets the spaces of $(p,q)$-forms be organised into a cochain complex. Once this is available, the language of closed forms, exact forms, and cohomology becomes meaningful for $\bar\partial$. Analytically, this identity is also the compatibility condition behind the equation $\bar\partial u=\alpha$. If $\alpha$ is not $\bar\partial$-closed, then it cannot be $\bar\partial u$ for any $u$. Thus the theorem tells us what the first possible obstruction is; the rest of the chapter studies when that only local obstruction is also sufficient globally. [/explanation] The identity $\bar\partial^2 = 0$ says that the sequence of spaces $\mathcal{E}^{p,0}(\Omega) \xrightarrow{\bar\partial} \mathcal{E}^{p,1}(\Omega) \xrightarrow{\bar\partial} \mathcal{E}^{p,2}(\Omega) \xrightarrow{\bar\partial} \cdots$ is a complex — the image of each $\bar\partial$ is contained in the kernel of the next. This is the **Dolbeault complex** $\mathcal{E}^{p,\bullet}(\Omega)$. A $(p,q)$-form $\alpha$ is called **$\bar\partial$-closed** if $\bar\partial \alpha = 0$, and **$\bar\partial$-exact** if $\alpha = \bar\partial \beta$ for some $\beta \in \mathcal{E}^{p,q-1}(\Omega)$. Since $\bar\partial^2 = 0$, every $\bar\partial$-exact form is $\bar\partial$-closed. A smooth function $f$ satisfies $\bar\partial f = 0$ precisely when $\frac{\partial f}{\partial \bar z_k} = 0$ for all $k$, which is exactly the system of Cauchy–Riemann equations: $f$ is holomorphic. So the holomorphic functions on $\Omega$ are precisely the elements of the kernel of $\bar\partial: \mathcal{E}^{p,0}(\Omega) \to \mathcal{E}^{p,1}(\Omega)$. The Dolbeault complex thus begins with holomorphic $p$-forms and measures, at each subsequent level, the failure of $\bar\partial$-exactness by $\bar\partial$-closed forms. ## The Dolbeault Complex and Cohomology The identity $\bar\partial^2 = 0$ means that $\bar\partial$-exact forms are always $\bar\partial$-closed, but the converse can fail globally — a form annihilated by $\bar\partial$ need not be the $\bar\partial$ of anything defined on all of $\Omega$. How do we measure this failure? The answer is cohomology: we take the quotient of closed forms by exact forms at each degree, and the resulting group records precisely the global obstruction to solving $\bar\partial u = \alpha$. [example: A Domain Where Global $\bar\partial$-Solvability Fails] Local solvability does not imply global solvability. The Hartogs figure \begin{align*} H = \{(z_1,z_2) \in \mathbb{D}^2 : |z_1| > 1/2\} \cup \{(z_1,z_2) \in \mathbb{D}^2 : |z_2| < 1/2\} \end{align*} is not pseudoconvex and is not a domain of holomorphy. This is the kind of domain where global $\bar\partial$-solvability should not be expected: holomorphic data on $H$ are constrained by forced extension to the larger bidisc, so local analytic data cannot always be patched freely. The cohomology groups defined below are designed to measure exactly this sort of global obstruction. By contrast, on Stein domains the global obstruction disappears: the vanishing theorem below gives $H^{p,q}_{\bar\partial}(\Omega)=0$ for $q \geq 1$. Later, once the Levi problem identifies pseudoconvex domains in $\mathbb{C}^n$ with Stein domains, this vanishing becomes available on the full pseudoconvex class. Thus Dolbeault cohomology is a detector of whether the domain has the right global shape for solving analytic patching problems. [/example] The example shows that the obstruction to solving $\bar\partial u = \alpha$ is not a defect of the notation but a genuine global phenomenon. To record that obstruction systematically, we compare all closed forms with the exact forms that actually arise as $\bar\partial$ of something one degree lower. The quotient is the Dolbeault cohomology group. [definition: Dolbeault Cohomology] The **Dolbeault cohomology group** of bidegree $(p,q)$ on $\Omega$ is \begin{align*} H^{p,q}_{\bar\partial}(\Omega) = \frac{\ker\!\left(\bar\partial: \mathcal{E}^{p,q}(\Omega) \to \mathcal{E}^{p,q+1}(\Omega)\right)}{\operatorname{im}\!\left(\bar\partial: \mathcal{E}^{p,q-1}(\Omega) \to \mathcal{E}^{p,q}(\Omega)\right)}. \end{align*} [/definition] The group $H^{p,q}_{\bar\partial}(\Omega)$ measures the global obstruction to solving the equation $\bar\partial u = \alpha$ for a given $\bar\partial$-closed $(p,q)$-form $\alpha$: the equation is solvable if and only if $\alpha$ represents the zero class in $H^{p,q}_{\bar\partial}(\Omega)$. The group $H^{p,0}_{\bar\partial}(\Omega)$ is always the space of holomorphic $p$-forms $\Omega^p(\Omega)$ on $\Omega$, since the kernel of $\bar\partial$ on $(p,0)$-forms is holomorphic $p$-forms and the image of $\bar\partial$ from $\mathcal{E}^{p,-1}$ is trivially zero. All the interesting cohomological information thus lives at degrees $q \geq 1$. Whether $H^{p,q}_{\bar\partial}(\Omega) = 0$ for $q \geq 1$ is the central question of the theory. Vanishing means that every $\bar\partial$-closed form of positive $(0,q)$ degree is $\bar\partial$-exact — that is, the $\bar\partial$-equation is always solvable. This is not automatic from the local result; it is a global property that depends on the topology and complex geometry of $\Omega$. ## The $\bar\partial$-Poincaré Lemma The Poincaré lemma in real differential geometry says that every closed form is locally exact. The $\bar\partial$-Poincaré lemma is the complex analogue: every $\bar\partial$-closed form is locally $\bar\partial$-exact. The proof, however, is substantially more involved than the real case because the $\bar\partial$-equation is a first-order overdetermined system that requires analysis to solve. The local solution is provided by the Cauchy–Green operator, which we first illustrate in one variable. [example: Solution of $\bar\partial u = f$ on a Disc via the Cauchy–Green Formula] Let $D \subset \mathbb{C}$ be the unit disc and let $f \in C^\infty(D)$ be a smooth function. We seek $u \in C^\infty(D)$ such that $\frac{\partial u}{\partial \bar z} = f$, i.e., $\bar\partial u = f\,d\bar z$. Define the **Cauchy–Green transform** \begin{align*} u(z) = \frac{1}{2\pi i} \int_D \frac{f(\zeta)}{\zeta - z}\, d\zeta \wedge d\bar\zeta. \end{align*} Writing $d\zeta \wedge d\bar\zeta = -2i\, d\xi\, d\eta$ (where $\zeta = \xi + i\eta$), this becomes \begin{align*} u(z) = \frac{1}{\pi} \int_D \frac{f(\zeta)}{\zeta - z}\, d\xi\, d\eta. \end{align*} This integral converges for each $z \in D$ since $1/|\zeta - z|$ is locally $L^1$ in $\mathbb{R}^2$. To verify that $\frac{\partial u}{\partial \bar z} = f$, one differentiates under the integral sign. The distributional identity $\frac{\partial}{\partial \bar z}\!\left(\frac{1}{\zeta - z}\right) = \pi\delta(\zeta - z)$ — the fundamental solution of $\bar\partial$ in one variable — gives \begin{align*} \frac{\partial u}{\partial \bar z}(z) = \frac{1}{\pi} \int_D f(\zeta)\, \pi\delta(\zeta - z)\, d\xi\, d\eta = f(z). \end{align*} The rigorous justification uses the Pompeiu formula: for $u \in C^1(\bar D)$, \begin{align*} u(z) = \frac{1}{2\pi i}\oint_{\partial D} \frac{u(\zeta)}{\zeta - z}\, d\zeta + \frac{1}{2\pi i}\int_D \frac{\bar\partial u(\zeta)}{\zeta - z}\, d\zeta \wedge d\bar\zeta, \end{align*} from which the statement $\frac{\partial}{\partial\bar z}(T f) = f$ follows by a limiting argument and Stokes's theorem. The Cauchy–Green transform therefore solves $\bar\partial u = f\,d\bar z$ on $D$, with $u$ smooth when $f$ is smooth. The solution is not unique: one may add any [holomorphic function](/page/Holomorphic%20Function) to $u$ and still have $\bar\partial u = f\,d\bar z$. [/example] The Cauchy–Green formula is the key ingredient in the $\bar\partial$-Poincaré lemma for higher-degree forms. In one variable the formula solves $\bar\partial u = f\,d\bar z$ on any disc, which is enough to conclude that every $\bar\partial$-closed $(0,1)$-form on a disc is $\bar\partial$-exact. The question is whether the same holds in several variables: given a $\bar\partial$-closed $(p,q)$-form on a polydisc with $q \geq 1$, is it always $\bar\partial$-exact? The strategy is to apply the one-variable Cauchy–Green formula coordinate by coordinate, peeling off one $d\bar z_k$ factor at each step in a process of successive integration. [quotetheorem:3410] [citeproof:3410] The theorem says that $H^{p,q}_{\bar\partial}(P) = 0$ for all $q \geq 1$ when $P$ is a polydisc. This is the analogue of the Poincaré lemma saying $H^k_{\mathrm{dR}}(U) = 0$ for contractible $U$: both say that locally there are no obstructions to solving the respective equations. The local vanishing does not imply global vanishing, just as the local Poincaré lemma does not imply that all closed forms on a given domain are exact. [remark: Grothendieck's Version] The version stated here is sometimes called the **Dolbeault–Grothendieck lemma**. Grothendieck's formulation emphasises that the polydisc $P$ can be replaced by any product of discs (not necessarily round), and that the result holds for forms with values in a holomorphic vector bundle — the proof applies verbatim because it is coordinate-by-coordinate and requires only the one-variable Cauchy–Green formula at each step. [/remark] ## Dolbeault Cohomology and the Cousin I Problem The Dolbeault cohomology $H^{p,q}_{\bar\partial}(\Omega)$ is defined analytically — as a quotient of spaces of differential forms — but what does it mean concretely for complex analysis? The question is whether a prescribed meromorphic singularity structure can be realised by a global [meromorphic function](/page/Meromorphic%20Function), and the answer turns out to depend on exactly one cohomology group. This connection to the Cousin I problem is the first indication that $H^{p,q}_{\bar\partial}(\Omega)$ is not a bookkeeping device but a genuine analytic invariant. The Cousin I problem on a domain $\Omega$ asks: given an open cover $\{U_i\}$ of $\Omega$ and meromorphic functions $f_i$ on $U_i$ satisfying $f_i - f_j \in \mathcal{O}(U_i \cap U_j)$ on each overlap, does there exist a global [meromorphic function](/page/Meromorphic%20Function) $f \in \mathcal{M}(\Omega)$ with $f - f_i \in \mathcal{O}(U_i)$ for all $i$? The datum $(f_i - f_j)$ is a Cousin I datum (or Weierstrass datum), and the functions $f_i - f_j$ are called the transition data. The connection to $\bar\partial$ cohomology is conceptual. The transition data of a Cousin I problem determine an obstruction class in the first cohomology of the sheaf of holomorphic functions, and Dolbeault's theorem identifies that obstruction with a class in $H^{0,1}_{\bar\partial}(\Omega)$. If the class vanishes, the local meromorphic data can be corrected by holomorphic terms and glued to a global [meromorphic function](/page/Meromorphic%20Function); if it does not vanish, the obstruction is genuine. The theorem below is the precise form of this equivalence. [quotetheorem:3387] [citeproof:3387] This equivalence makes Dolbeault cohomology a functional tool, not merely an algebraic formality. The question of whether meromorphic functions with prescribed poles exist — a classical question of the theory of functions — reduces to the vanishing of a single cohomology group. We will return to the Cousin problems more carefully in Chapter 7. [remark: When $H^{0,1}_{\bar\partial}(\Omega) \neq 0$ and the Cousin II Caveat] If $H^{0,1}_{\bar\partial}(\Omega) \neq 0$, then there exists a Cousin I datum on $\Omega$ that cannot be solved: no global [meromorphic function](/page/Meromorphic%20Function) with the prescribed polar behaviour exists. The non-trivial cohomology class is the obstruction. This happens, for example, on the Hartogs figure $\{(z_1,z_2) \in \mathbb{D}^2 : |z_1| > 1/2\} \cup \{(z_1,z_2) \in \mathbb{D}^2 : |z_2| < 1/2\}$, which is not pseudoconvex and has $H^{0,1}_{\bar\partial} \neq 0$. The Cousin II problem — prescribing a divisor (zeros and poles with multiplicities) rather than just poles — is harder. Even when $H^{0,1}_{\bar\partial}(\Omega) = 0$ and Cousin I is always solvable, Cousin II can fail. Its solvability requires the vanishing of $H^{0,1}_{\bar\partial}(\Omega)$ together with a condition on the first Chern class of the line bundle associated to the divisor; this is a topological obstruction that $\bar\partial$-cohomology alone does not see. Cousin II is discussed in Chapter 7. [/remark] [example: Cousin I Data on a Polydisc] Let $P = \mathbb{D}^2$ be the bidisc. A simple way to see the bookkeeping in the definition is to start from a global meromorphic function and restrict it to an open cover. Cover $P$ by three open sets: \begin{align*} U_1 &= \{z_1 \neq 0\}, \quad U_2 = \{z_2 \neq 0\}, \quad U_3 = \{|z_1|<1/2,\ |z_2|<1/2\}, \end{align*} where each set is understood as a subset of $P$. These three sets cover $P$. Define a meromorphic function $F(z_1,z_2)=1/z_1$ on $P$, and set \begin{align*} f_i = F|_{U_i}. \end{align*} Then $f_i-f_j=0$ on every overlap, so this is a Cousin I datum with zero transition functions. The global solution is $F$ itself. This example is deliberately elementary: it shows what local meromorphic representatives and holomorphic transition functions look like without hiding the definition behind cohomology. The theorem guarantees much more on the polydisc, namely that every compatible Cousin I datum, not only one obtained by restriction from a known global function, has a global meromorphic solution. [/example] ## Vanishing of Dolbeault Cohomology on Stein Manifolds The $\bar\partial$-Poincaré lemma gives vanishing of Dolbeault cohomology locally — on any polydisc. Globally, the cohomology $H^{p,q}_{\bar\partial}(X)$ can be non-zero; its vanishing on a whole complex manifold $X$ is a deep global result. The theorem quoted here is deliberately stated for Stein manifolds, because the equivalence between pseudoconvex domains and Stein domains is the Levi problem and has not yet been proved in this narrative. For the purposes of this course, a **Stein manifold** is a complex manifold on which the familiar function theory of domains of holomorphy works globally: holomorphic functions separate points, compact sets have compact holomorphic hulls, and global holomorphic functions are abundant enough for approximation and cohomology vanishing. Every domain of holomorphy in $\mathbb{C}^n$ is Stein, and after the Levi problem is solved, pseudoconvex domains in $\mathbb{C}^n$ will be seen to be Stein as well. The fundamental theorem, due to Hörmander (1965) and proved using $L^2$ methods, is: [quotetheorem:3388] [citeproof:3388] The theorem is the analytic engine behind global several-variable function theory on Stein spaces. A Stein exhaustion supplies the positivity needed to control solutions of the $\bar\partial$-equation; in schematic form, Hörmander's estimates compare a solution $u$ with the input form $\alpha$ using weighted $L^2$ norms. The precise functional-analytic construction belongs to Course III. Here the conceptual point is enough: Stein geometry turns local $\bar\partial$-solvability into global $\bar\partial$-solvability. The immediate consequence is that on a Stein manifold, the Cousin I problem is always solvable, and more generally every $\bar\partial$-closed $(p,q)$-form has a $\bar\partial$-primitive for $q \geq 1$. For domains in $\mathbb{C}^n$, this result will later be combined with the Levi problem to recover the familiar pseudoconvex-domain formulation. Until that equivalence is established, we use the theorem only in its Stein form. [explanation: Why This Vanishing Theorem Matters] The vanishing theorem $H^{p,q}_{\bar\partial}(X) = 0$ for $q \geq 1$ on Stein manifolds is the cohomological expression of the fact that Stein spaces are the natural global setting for holomorphic approximation, Cousin problems, and $\bar\partial$-solvability. To see the contrast, consider what happens on a non-pseudoconvex domain. The Hartogs figure $\mathbb{H} = \{(z_1,z_2) \in \mathbb{D}^2 : |z_1| > 1/2\} \cup \{(z_1,z_2) \in \mathbb{D}^2 : |z_2| < 1/2\}$ is not pseudoconvex (Chapter 5). Its failure is visible function-theoretically: every [holomorphic function](/page/Holomorphic%20Function) on $\mathbb{H}$ extends to the strictly larger bidisc, so $\mathbb{H}$ cannot be a domain of holomorphy. Later, after the Levi problem is available, this same contrast can be read cohomologically as the failure of the Stein-type vanishing above. Conversely, the psh exhaustion function on a pseudoconvex domain later plays a dual role: it is the global geometric datum and, in the $L^2$ theory, the source of the weights used to solve $\bar\partial$. This is the mechanism that eventually translates "geometric condition on the domain" into "analytic solvability of $\bar\partial$." The full $L^2$ theory is the subject of Course III; in this course, the point is to see how the Stein vanishing theorem and the Levi problem fit together. [/explanation] ## The Dolbeault Isomorphism The analytic definition of $H^{p,q}_{\bar\partial}(\Omega)$ depends on smooth $(p,q)$-forms and the $\bar\partial$ operator, yet the problems one wants to solve — Cousin I, the Levi problem — are phrased in holomorphic data: meromorphic functions, holomorphic transitions, sheaves of holomorphic forms. How do the two perspectives relate? Are they computing the same invariants, or does the passage to smooth forms introduce extraneous information? The [Dolbeault isomorphism](/theorems/3389) answers this: the analytic Dolbeault groups coincide with the sheaf cohomology groups of the holomorphic forms sheaf, computed via Čech theory with holomorphic data. Let $\Omega^p = \Omega^p_\Omega$ denote the sheaf of holomorphic $p$-forms on $\Omega$. The theorem below says that the Dolbeault complex is not merely an analytic gadget: it computes the same obstruction groups as the sheaf $\Omega^p$. The local reason is the $\bar\partial$-Poincaré lemma, and the global reason belongs to the proof of the theorem rather than to the surrounding notes. [quotetheorem:3389] [citeproof:3389] The theorem identifies two languages for the same obstruction. Smooth forms and the operator $\bar\partial$ give an analytic model; holomorphic forms and sheaf cohomology give an algebraic model. Chapter 7 uses this dictionary without needing to rebuild the proof of the dictionary each time. The [Dolbeault isomorphism](/theorems/3389) is important because it allows one to use either the analytic definition (solving $\bar\partial u = \alpha$) or the sheaf-theoretic definition (Čech cohomology with holomorphic forms) interchangeably, and both give the same groups. For computation, the analytic side is often more tractable: one can use $L^2$ estimates and PDE methods. For structural results — functoriality, base change, comparison with other cohomology theories — the sheaf-theoretic side is more powerful. In Chapter 7, the isomorphism is used in both directions: analytic vanishing (from Hörmander) implies sheaf-theoretic vanishing (Oka's theorem), and the sheaf picture organises the Cousin problems into a coherent framework. In particular, the Stein vanishing theorem translates via Dolbeault to sheaf-cohomology vanishing for holomorphic forms on Stein spaces. After the Levi problem identifies pseudoconvex domains in $\mathbb{C}^n$ with Stein domains, this same mechanism becomes one of the main inputs for the pseudoconvex-domain formulation used in Chapter 8. [example: Punctures and Additive Cohomology] Punctures are a useful warning about which obstruction is being measured. The punctured plane $\mathbb{C}\setminus\{0\}$ has non-trivial topology, but it is an open Riemann surface and hence has vanishing additive Cousin cohomology: \begin{align*} H^{0,1}_{\bar\partial}(\mathbb{C}\setminus\{0\}) \cong H^1(\mathbb{C}\setminus\{0\},\mathcal{O}) = 0. \end{align*} For example, the form $(1/\bar z)\,d\bar z$ is $\bar\partial$-exact on the punctured plane because $2\log|z|$ is smooth there and $\bar\partial(2\log|z|)=(1/\bar z)\,d\bar z$. In two variables the Hartogs phenomenon changes the story. The domain $\mathbb{C}^2\setminus\{0\}$ is not a domain of holomorphy: every holomorphic function on it extends to all of $\mathbb{C}^2$. Correspondingly, $H^1(\mathbb{C}^2\setminus\{0\},\mathcal{O})$ is non-zero, and the Cousin datum from Chapter 2 records this obstruction. Thus the punctured plane and the punctured space have opposite behaviour for different reasons: in one variable the puncture supports functions like $1/z$, while in two variables an isolated puncture is removable for holomorphic functions. [/example] ## Summary The $\bar\partial$ operator on $(p,q)$-forms satisfies $\bar\partial^2 = 0$, making the Dolbeault complex $\mathcal{E}^{p,\bullet}(\Omega)$ a cochain complex. Its cohomology groups $H^{p,q}_{\bar\partial}(\Omega)$ measure the global obstruction to solving $\bar\partial u = \alpha$ for $\bar\partial$-closed $(p,q)$-forms $\alpha$. The $\bar\partial$-Poincaré lemma says $H^{p,q}_{\bar\partial}(P) = 0$ for all $q \geq 1$ on any polydisc $P$, giving local exactness. Globally, $H^{0,1}_{\bar\partial}(\Omega) = 0$ is equivalent to the solvability of every Cousin I problem on $\Omega$, connecting $\bar\partial$ cohomology directly to the classical problem of prescribing poles. The formal global vanishing theorem in this chapter is stated for Stein manifolds; after the Levi problem is solved, pseudoconvex domains in $\mathbb{C}^n$ enter this class. The [Dolbeault isomorphism](/theorems/3389) $H^{p,q}_{\bar\partial}(\Omega) \cong H^q(\Omega, \Omega^p)$ connects the analytic and sheaf-theoretic perspectives, and will be used extensively in Chapters 7 and 8. --- The Cousin problems ask whether compatible local meromorphic or divisor data can be patched together globally. Domains of holomorphy provide the analytic setting where the additive problem is solvable, while the multiplicative problem also remembers topology through line bundles and Chern classes. # 7. The Cousin Problems The Cousin problems are the several-variable analogues of classical interpolation and factorisation problems from one complex variable. In one variable, the [Mittag-Leffler theorem](/theorems/3367) guarantees a [meromorphic function](/page/Meromorphic%20Function) with any prescribed principal parts, and the Weierstrass product theorem realises any divisor by an entire function. The natural question — do these theorems persist in $\mathbb{C}^n$ for $n \geq 2$? — has a nuanced answer: the additive (Cousin I) problem is always solvable on domains of holomorphy, but the multiplicative (Cousin II) problem requires additional topological conditions. Chapter 6 established that the vanishing of $H^{0,1}_{\bar\partial}(\Omega)$ is equivalent to Cousin I solvability via the [Dolbeault isomorphism](/theorems/3389). Here we make this connection precise in sheaf-theoretic language, prove [Oka's theorem on Cousin I](/theorems/3411), develop the Cousin II obstruction through the exponential sequence, and establish the Weierstrass product theorem in $\mathbb{C}^n$ as a corollary. ## The Additive Cousin Problem The [Mittag-Leffler theorem](/theorems/3367) in one variable says: given an open cover $\{U_i\}$ of a domain in $\mathbb{C}$ and meromorphic functions $f_i$ on each $U_i$ whose differences $f_i - f_j$ are holomorphic on $U_i \cap U_j$, there exists a global [meromorphic function](/page/Meromorphic%20Function) $f$ on the whole domain such that $f - f_i$ is holomorphic on each $U_i$. The Cousin I problem asks whether the same holds in several variables, and the answer depends on the geometry of the domain. The underlying difficulty is that specifying local meromorphic data compatibly does not automatically produce a global [meromorphic function](/page/Meromorphic%20Function). Two patches $U_1, U_2$ covering $\Omega$ might carry meromorphic functions $f_1, f_2$ with $f_1 - f_2$ holomorphic on the overlap, yet there may be no $f \in \mathcal{M}(\Omega)$ restricting to $f_1$ and $f_2$ up to holomorphic corrections. The failure is measured by a cohomology class. [definition: Cousin I Datum] Let $\Omega \subset \mathbb{C}^n$ be a domain and $\{U_i\}_{i \in I}$ an open cover of $\Omega$. A **Cousin I datum** on this cover is a collection of meromorphic functions $f_i \in \mathcal{M}(U_i)$ such that \begin{align*} g_{ij} := f_i - f_j \in \mathcal{O}(U_i \cap U_j) \end{align*} for every pair $i, j$ with $U_i \cap U_j \neq \varnothing$. A **solution** to the Cousin I problem for this datum is a [meromorphic function](/page/Meromorphic%20Function) $f \in \mathcal{M}(\Omega)$ satisfying $f - f_i \in \mathcal{O}(U_i)$ for every $i \in I$. [/definition] The holomorphic differences $g_{ij}$ automatically satisfy the cocycle condition: on any triple overlap $U_i \cap U_j \cap U_k$, \begin{align*} g_{ij} + g_{jk} + g_{ki} = (f_i - f_j) + (f_j - f_k) + (f_k - f_i) = 0. \end{align*} The collection $(g_{ij})$ is therefore a Čech 1-cocycle with values in the sheaf $\mathcal{O}$. If a global solution $f$ exists and we set $h_i = f - f_i \in \mathcal{O}(U_i)$, then $g_{ij} = f_i - f_j = (f - h_i) - (f - h_j) = h_j - h_i$, so the cocycle is a coboundary. The converse holds too: if $g_{ij} = h_j - h_i$ for some $h_i \in \mathcal{O}(U_i)$, then $f_i + h_i = f_j + h_j$ on every overlap, so these local expressions define a global [meromorphic function](/page/Meromorphic%20Function). The obstruction is now visible: the local principal parts fail to glue precisely when this cocycle represents a nonzero cohomology class. Thus the problem is no longer to guess a global meromorphic function directly, but to decide whether the holomorphic discrepancies on overlaps can be absorbed by changing the local representatives. [quotetheorem:3390] [citeproof:3390] This is an if-and-only-if: the cohomology group $H^1(\Omega, \mathcal{O})$ captures the precise obstruction. The [Dolbeault isomorphism](/theorems/3389) from Chapter 6 identifies $H^1(\Omega, \mathcal{O}) \cong H^{0,1}_{\bar\partial}(\Omega)$, so vanishing of Dolbeault cohomology is equivalent to Cousin I solvability. In this narrative, that vanishing has been stated first for Stein manifolds; after the Levi problem is available, pseudoconvex domains in $\mathbb{C}^n$ enter the same vanishing framework. [example: Cousin I on $\mathbb{C}^2$ with Prescribed Polar Set] Let $\Omega = \mathbb{C}^2$ and cover it by \begin{align*} U_0 &= \{(z_1,z_2): |z_1|^2+|z_2|^2<1\},\\ U_1 &= \{z_1 \neq 0\},\\ U_2 &= \{z_2 \neq 0\}. \end{align*} These three open sets cover $\mathbb{C}^2$: the only point missed by $U_1\cup U_2$ is the origin, which lies in $U_0$. Let \begin{align*} F=\frac{1}{z_1}+\frac{1}{z_2}\in \mathcal{M}(\mathbb{C}^2), \end{align*} and set $f_i=F|_{U_i}$ for $i=0,1,2$. On every overlap the difference is \begin{align*} f_i-f_j=0, \end{align*} which is holomorphic. This is a valid Cousin I datum, and the global meromorphic function $F$ itself solves it. The example is intentionally transparent: it records how a global polar set is encoded by local meromorphic representatives, while Oka's theorem is needed for compatible data that are not already presented as restrictions of a known global meromorphic function. [/example] ## Oka's Theorem on Cousin I The cohomological criterion reduces Cousin I solvability to the vanishing of $H^1(\Omega, \mathcal{O})$. The next question is: on which spaces does this vanishing hold? The theorem below is stated in Stein language: on a Stein manifold, additive Cousin data are solvable. For domains of holomorphy in $\mathbb{C}^n$, one applies this through the standard identification of such domains with Stein domains. [quotetheorem:3411] [citeproof:3411] Oka's theorem is a structural result of the first order: the Stein condition forces global Cousin I solvability, and domains of holomorphy inherit this conclusion when they are viewed as Stein domains. Conceptually it is one place where geometry, cohomology, and the $\bar\partial$-equation meet. The [solution of the Levi problem](/theorems/3416) in Chapter 8 will explain why pseudoconvexity is the geometric condition that supports this vanishing theory. The converse direction is more delicate: Cousin I solvability on $\Omega$ does not automatically imply $\Omega$ is a domain of holomorphy, without additional hypotheses. But once the Levi problem is solved (Chapter 8), one has the full circle: domain of holomorphy $\Leftrightarrow$ pseudoconvex $\Rightarrow H^{0,1}_{\bar\partial} = 0 \Rightarrow$ Cousin I solvable, and the Cousin problems are resolved completely on pseudoconvex domains. ## The Multiplicative Cousin Problem Cousin I controls only the polar structure of a [meromorphic function](/page/Meromorphic%20Function), up to holomorphic corrections. A harder problem — prescribing both zero and pole loci with multiplicities — requires the Cousin II framework. Even on a domain where Cousin I is always solvable, Cousin II can fail: the additional obstruction is topological, not analytic. [definition: Cousin II Datum] Let $\Omega \subset \mathbb{C}^n$ be a domain and $\{U_i\}_{i \in I}$ an open cover. A **Cousin II datum** is a collection of meromorphic functions $f_i \in \mathcal{M}^*(U_i)$ — meromorphic and not identically zero on any component — such that \begin{align*} g_{ij} := f_i / f_j \in \mathcal{O}^*(U_i \cap U_j) \end{align*} for every $i, j$ with $U_i \cap U_j \neq \varnothing$, where $\mathcal{O}^*(V)$ denotes the group of nowhere-vanishing holomorphic functions on $V$. A **solution** is $f \in \mathcal{M}^*(\Omega)$ such that $f / f_i \in \mathcal{O}^*(U_i)$ for every $i \in I$. [/definition] The ratios $g_{ij} = f_i/f_j$ form a Čech 1-cocycle for the sheaf $\mathcal{O}^*$: on triple overlaps, $g_{ij} g_{jk} g_{ki} = (f_i/f_j)(f_j/f_k)(f_k/f_i) = 1$. Here the obstruction is multiplicative rather than additive, so it naturally lives in cohomology with coefficients in nowhere-vanishing holomorphic functions. The theorem below gives the Cousin II analogue of the Cousin I criterion and identifies exactly what must vanish for the local divisors to glue. [quotetheorem:3391] [citeproof:3391] The shift from Cousin I to Cousin II changes the obstruction group. Additive transition functions live in the sheaf $\mathcal{O}$, while multiplicative transition functions live in $\mathcal{O}^*$ and define a holomorphic line bundle. Thus a Cousin II datum is solvable exactly when its associated line bundle is holomorphically isomorphic to the product bundle. Here $\mathcal{O}(U)$ denotes holomorphic functions on $U$, $\mathcal{O}^*(U)$ denotes nowhere-vanishing holomorphic functions, $\mathcal{M}(U)$ denotes meromorphic functions, and $\mathcal{M}^*(U)$ denotes nonzero meromorphic functions. A sheaf is a rule assigning such data to each open set and allowing compatible local data to glue. The group $H^1(\Omega,\mathcal{O}^*)$, also called the Picard group $\operatorname{Pic}(\Omega)$, classifies holomorphic line bundles on $\Omega$; the first Chern class $c_1$ records the topological obstruction carried by such a line bundle. The relationship between Cousin I and Cousin II is captured by the **exponential sheaf sequence** \begin{align*} 0 \to \underline{\mathbb{Z}} \xrightarrow{\; \cdot 2\pi i \;} \mathcal{O} \xrightarrow{\;\exp\;} \mathcal{O}^* \to 0, \end{align*} where $\underline{\mathbb{Z}}$ is the constant sheaf of integers, and $\exp(g) = e^{2\pi i g}$. This is an exact sequence of sheaves (the exponential map is locally surjective because every nowhere-vanishing [holomorphic function](/page/Holomorphic%20Function) has a local logarithm). Applying the long exact cohomology sequence gives \begin{align*} \cdots \to H^1(\Omega, \mathbb{Z}) \to H^1(\Omega, \mathcal{O}) \to H^1(\Omega, \mathcal{O}^*) \xrightarrow{c_1} H^2(\Omega, \mathbb{Z}) \to H^2(\Omega, \mathcal{O}) \to \cdots \end{align*} The map $c_1: \operatorname{Pic}(\Omega) \to H^2(\Omega, \mathbb{Z})$ is the **first Chern class** of the line bundle. From this sequence one reads off the crucial implication: if $H^1(\Omega, \mathcal{O}) = 0$ (Cousin I always solvable) and $H^2(\Omega, \mathbb{Z}) = 0$ (no topological obstruction), then $H^1(\Omega, \mathcal{O}^*) = 0$ and Cousin II is always solvable. The first condition is guaranteed on domains of holomorphy by Oka's theorem; the second is a topological condition that may or may not hold. [quotetheorem:3412] [citeproof:3412] The hypothesis $H^2(\Omega, \mathbb{Z}) = 0$ is genuinely necessary. When $H^2(\Omega, \mathbb{Z}) \neq 0$, nontorsion line bundles exist on $\Omega$, and Cousin II can fail even though Cousin I is always solvable. [example: The Topological Obstruction on $(\mathbb{C}^*)^2$] The domain $\Omega = (\mathbb{C}^*)^2 = \{(z_1, z_2) \in \mathbb{C}^2 : z_1 z_2 \neq 0\}$ is a Stein manifold (a domain of holomorphy in the generalised sense), so $H^1(\Omega, \mathcal{O}) = 0$ and every Cousin I datum on $\Omega$ is solvable. However, $\Omega$ is homotopy equivalent to the 2-torus $T^2 = S^1 \times S^1$, so \begin{align*} H^2(\Omega, \mathbb{Z}) \cong H^2(T^2, \mathbb{Z}) \cong \mathbb{Z} \neq 0. \end{align*} In the analytic category, the exponential sequence and Stein vanishing identify the holomorphic Picard group with $H^2(\Omega,\mathbb{Z})$. Thus the holomorphic Picard group is nonzero. This is a holomorphic statement; it should not be confused with the algebraic Picard group of the affine algebraic torus, which is trivial. Consequently there are holomorphic line bundles on $(\mathbb{C}^*)^2$ with nonzero first Chern class. Any Cousin II datum whose transition functions define such a line bundle cannot be solved by a single global meromorphic function: a solution would trivialise the line bundle, forcing its first Chern class to vanish. This is the clean lesson of the example: Cousin I has no obstruction on this Stein domain, but Cousin II still sees the topology of the underlying torus. [/example] ## Divisors and the Weierstrass Product Theorem in $\mathbb{C}^n$ In one complex variable, the Weierstrass product theorem realises any discrete zero set with prescribed multiplicities as the zero set of an entire function. The natural analogue in several variables concerns analytic hypersurfaces, and Cousin II is exactly the right tool. [definition: Divisor] Let $\Omega \subset \mathbb{C}^n$ be an [open set](/page/Open%20Set). A **divisor** on $\Omega$ is a locally finite formal sum \begin{align*} D = \sum_{j} n_j V_j, \end{align*} where each $V_j$ is an irreducible analytic hypersurface in $\Omega$, each $n_j \in \mathbb{Z} \setminus \{0\}$, and locally finite means every compact $K \subset \Omega$ meets only finitely many $V_j$. The **divisor of a [meromorphic function](/page/Meromorphic%20Function)** $f \in \mathcal{M}(\Omega)$ is $\operatorname{div}(f) = \sum_V \operatorname{ord}_V(f) \cdot V$, where $\operatorname{ord}_V(f)$ is the vanishing order of $f$ along $V$ (positive for zeros, negative for poles, zero when $f$ does not vanish or have a pole along $V$). A divisor $D$ is **effective** if all $n_j > 0$. [/definition] Every [meromorphic function](/page/Meromorphic%20Function) determines a divisor. The Weierstrass question is whether every divisor arises from a [meromorphic function](/page/Meromorphic%20Function). Locally, this is always true: by the [Weierstrass preparation theorem](/theorems/3381) (Chapter 1), every irreducible analytic hypersurface in a neighbourhood of any point is locally defined by a single [holomorphic function](/page/Holomorphic%20Function), so a divisor locally defines a Cousin II datum. The obstruction to a global realisation is precisely $H^1(\Omega, \mathcal{O}^*)$. [quotetheorem:3413] [citeproof:3413] The classical Weierstrass product theorem in one variable follows as a special case: a locally finite collection of points $\{a_k\} \subset \mathbb{C}$ with multiplicities $n_k$ defines an effective divisor on $\mathbb{C}$, and the theorem produces an entire function $f$ with $f(a_k) = 0$ to order $n_k$ and no other zeros. In one variable the local Weierstrass polynomials are simply linear factors $(z - a_k)^{n_k}$, and the product may need Weierstrass elementary factors for convergence — the Cousin II machinery subsumes this convergence argument. [remark: Local vs. Global in the Weierstrass Theorem] The theorem separates two kinds of information. Locally, Weierstrass preparation turns analytic hypersurfaces into zero sets of one-variable polynomials with holomorphic coefficients. Globally, the question is whether those local defining functions can be made compatible on overlaps; the obstruction is measured by $H^1(\Omega, \mathcal{O}^*)$. This separation of local algebra from global geometry is one of the principal insights of the sheaf-theoretic approach to several complex variables. [/remark] ## Relation to the Classical Mittag-Leffler and Weierstrass Theorems The Cousin machinery clarifies exactly why the classical one-variable theorems hold and what makes the several-variable setting harder. The **[Mittag-Leffler theorem](/theorems/3367)** in one variable is Cousin I on domains in $\mathbb{C}$: given compatible local meromorphic prescriptions, find a global solution. In cohomological language, its validity on open Riemann surfaces corresponds to the vanishing of $H^1(\Omega,\mathcal{O})$. That vanishing is a global one-dimensional theorem, not a consequence of local exactness alone. In several variables, $H^1(\Omega, \mathcal{O})$ can be non-zero when $\Omega$ is not pseudoconvex, and Cousin I can fail. The **Weierstrass product theorem** in one variable is Cousin II on domains in $\mathbb{C}$: every divisor (discrete set with multiplicities) is the divisor of a [meromorphic function](/page/Meromorphic%20Function). This holds on every domain in $\mathbb{C}$ because $H^2(\Omega, \mathbb{Z}) = 0$ for open sets $\Omega \subset \mathbb{C}$ (open subsets of $\mathbb{C}$ are homotopy equivalent to graphs, with vanishing $H^2$). In several variables, $H^2(\Omega, \mathbb{Z})$ can be nonzero, and Cousin II can fail even when Cousin I is always solvable. The one-variable theorems thus hold because the topology of open sets in $\mathbb{C}$ is simple enough that all obstructions vanish automatically. In several variables, the topology of $\Omega$ is an independent invariant that can obstruct global solutions even when all local data is compatible. [example: A Cousin I Datum with No Global Solution on a Non-Pseudoconvex Domain] The Hartogs figure $\mathbb{H} = \{(z_1, z_2) \in \mathbb{D}^2 : |z_1| > 1/2\} \cup \{(z_1, z_2) \in \mathbb{D}^2 : |z_2| < 1/2\}$ is not a domain of holomorphy and is not pseudoconvex (as established in Chapter 2); consequently $H^1(\mathbb{H}, \mathcal{O}) \neq 0$. The failure of Cousin I on $\mathbb{H}$ arises because every [holomorphic function](/page/Holomorphic%20Function) on $\mathbb{H}$ extends holomorphically to the full bidisc $\mathbb{D}^2$ by the [Hartogs extension theorem](/theorems/3401) (Chapter 2). Any putative global meromorphic solution $f$ to a Cousin I datum on $\mathbb{H}$ would also extend to $\mathbb{D}^2$. The extended function on $\mathbb{D}^2$ would then be forced to satisfy constraints incompatible with the original local datum, producing a contradiction. Cousin I data on $\mathbb{H}$ whose transition functions encode a nonzero class in $H^1(\mathbb{H}, \mathcal{O})$ have no solution because any solution would extend and thereby force the class to be zero. [/example] ## The Unified Sheaf-Cohomological Picture The Cousin problems, sheaf cohomology, and the geometry of domains of holomorphy all fit into a single coherent framework. A *coherent analytic sheaf* is, roughly, a sheaf of $\mathcal{O}$-modules that is locally described by finitely many holomorphic generators and finitely many holomorphic relations; examples include $\mathcal{O}$ itself, ideal sheaves of analytic subsets, and sheaves of sections of holomorphic vector bundles. On a domain of holomorphy $\Omega$, the general vanishing theorem — Cartan's Theorem B — states that $H^q(\Omega, \mathcal{F}) = 0$ for every coherent analytic sheaf $\mathcal{F}$ and every $q \geq 1$. [quotetheorem:3414] [citeproof:3414] [Oka's theorem on Cousin I](/theorems/3411) is the special case $q = 1$, $\mathcal{F} = \mathcal{O}$. Cartan's Theorem B in full generality subsumes all the cohomological vanishing results of the theory: it is the statement that domains of holomorphy are the "correct" setting for global problems in several complex variables. The proof uses either $L^2$ methods (Hörmander) or the machinery of coherent sheaves and the Oka–Cartan theory; it will appear in Course III. The exponential sequence, together with Theorem B, pins down the Picard group completely: for a domain of holomorphy, $H^1(\Omega, \mathcal{O}) = 0$ and $H^2(\Omega, \mathcal{O}) = 0$, so the long exact sequence collapses to \begin{align*} 0 \to H^1(\Omega, \mathcal{O}^*) \xrightarrow{c_1} H^2(\Omega, \mathbb{Z}) \to 0, \end{align*} giving the isomorphism $\operatorname{Pic}(\Omega) \cong H^2(\Omega, \mathbb{Z})$. The Picard group of a domain of holomorphy is a purely topological invariant. This is an instance of the **Oka principle**: on domains of holomorphy (and more generally Stein manifolds), analytic classification problems for holomorphic objects reduce to topological ones once the coherent-sheaf cohomology groups vanish. Cousin I solvability is controlled by $H^1(\Omega,\mathcal{O})$, which vanishes on domains of holomorphy by Oka's theorem; Cousin II solvability additionally requires the relevant class in $H^2(\Omega, \mathbb{Z})$ to vanish; and holomorphic vector bundles are classified by their underlying topological data. The complex-analytic world on Stein domains is topologically constrained in a precise way. This perspective points forward to Chapter 8: the Levi problem — proving that pseudoconvex $\Rightarrow$ domain of holomorphy — completes the cycle of equivalences and validates the Oka principle on the largest natural class of domains. Once the Levi problem is solved, Cartan's Theorem B applies to all pseudoconvex domains, and the Cousin problems are resolved in full generality. --- The Cousin and sheaf-theoretic results explain why cohomology vanishes on domains of holomorphy. The next step is the Levi problem: starting from the geometric condition of pseudoconvexity, prove that the domain actually has enough holomorphic functions to be a domain of holomorphy. # 8. Solution of the Levi Problem The Levi problem is the deepest question of the course, and this chapter is where it is finally answered. Posed by Levi in 1910, the problem asks whether pseudoconvexity — a condition detected entirely at the boundary via the Levi form — is equivalent to being a domain of holomorphy, a condition about the global behaviour of holomorphic functions. The answer is yes, and the proof, completed by Oka in 1942 and independently confirmed by Bremermann and Norguet in 1954, draws on virtually every tool developed in the preceding chapters: holomorphic convexity from Chapter 3, plurisubharmonic functions from Chapter 4, pseudoconvexity from Chapter 5, the $\bar\partial$-equation from Chapter 6, and the Cousin problems from Chapter 7. The full equivalence theorem is the capstone of the classical theory. ## Historical Context and the Statement of the Problem Levi formulated the problem in 1910, asking whether the geometric condition at the boundary that he had identified — positive semidefiniteness of the restricted complex Hessian of the defining function — forces a domain to be one on which holomorphic functions can develop genuine singularities. His question was sharpened by the [Cartan–Thullen theorem](/theorems/3385) (Chapter 3): a domain is a domain of holomorphy if and only if it is holomorphically convex, and this equivalence recasts the Levi problem as asking whether the geometric condition of pseudoconvexity is equivalent to the analytic condition of holomorphic convexity. The difficulty is deep. Pseudoconvexity is stated in terms of plurisubharmonic functions — real-valued functions satisfying a differential inequality — while holomorphic convexity requires the existence of genuinely holomorphic functions that separate points from the holomorphic hull. There is no direct algebraic mechanism for passing from the existence of a psh exhaustion to the existence of holomorphic functions with prescribed growth. Oka's breakthrough was to identify an indirect route: use the psh exhaustion to control an increasing family of compact pieces, solve local patching and approximation problems on those pieces, and then pass to global holomorphic functions by a limiting argument. ## Oka's Strategy: Blowing Up at Boundary Points The core of Oka's approach is the following idea. To show that a pseudoconvex domain $\Omega$ is a domain of holomorphy, it suffices to show that for every boundary point $p \in \partial\Omega$, there exists a [holomorphic function](/page/Holomorphic%20Function) $f \in \mathcal{O}(\Omega)$ that is unbounded near $p$. A domain of holomorphy requires a single function that cannot be continued across any boundary point simultaneously, but individual blow-up functions can be combined into a single non-extendable function by taking a convergent series of suitably normalised summands. The construction of a function blowing up at a given boundary point uses the pseudoconvexity assumption in a precise way. The exhaustion function $\phi$ satisfies $\phi(z) \to +\infty$ as $z \to \partial\Omega$, so the sublevel sets $\Omega_c = \{\phi < c\}$ exhaust $\Omega$ by relatively compact pieces. Oka's method turns control on these pieces into global holomorphic functions without assuming in advance that $\Omega$ is already a domain of holomorphy. The approximation theorem below is one of the tools that explains why holomorphic convexity, once obtained, is so powerful. ## The Oka–Weil Approximation Theorem Holomorphic functions on a domain of holomorphy enjoy a strong approximation property: any [holomorphic function](/page/Holomorphic%20Function) defined only on a compact piece of the domain can be shadowed arbitrarily closely by functions defined globally on all of $\Omega$. In the language used by the theorem card, a *Stein compact* is a compact set that has arbitrarily small Stein neighbourhoods. In the domain case, the parallel notion to keep in mind is a holomorphically convex compact set — a compact $K$ that already contains its own holomorphic hull. Without this hypothesis the conclusion fails: if $K$ is not holomorphically convex, there exist points in $\hat{K}_\Omega \setminus K$ at which a global function may take values not constrained by the supremum on $K$, so uniform approximation on $K$ alone cannot force the correct behaviour at those hull points. Recall from Chapter 3 that the holomorphic hull of a compact set $K \Subset \Omega$ is \begin{align*} \hat{K}_\Omega = \{ z \in \Omega : |f(z)| \leq \sup_K |f| \text{ for all } f \in \mathcal{O}(\Omega) \}. \end{align*} The set $\hat{K}_\Omega$ is always closed in $\Omega$ and contains $K$, but need not be compact: compactness of the holomorphic hull for every $K \Subset \Omega$ is precisely the condition of holomorphic convexity (Chapter 3). For a domain that is not a domain of holomorphy, the hull can accumulate at $\partial\Omega$. When $\Omega$ is a domain of holomorphy, the following theorem converts this structural information into an approximation statement. [quotetheorem:3415] [citeproof:3415] The Oka–Weil theorem is the several-variable analogue of Runge's theorem: in one complex variable, any [holomorphic function](/page/Holomorphic%20Function) on a compact subset $K$ of a simply connected domain can be approximated uniformly on $K$ by functions holomorphic on the whole domain. In several variables, simple connectivity is replaced by holomorphic convexity, which is the correct notion. The theorem illustrates the payoff of holomorphic convexity: once compact hulls are under control, global holomorphic functions are plentiful enough to approximate local holomorphic data on compact sets. ## The Full Equivalence Theorem The preceding results leave four apparently different tests for the same geometric phenomenon: enough holomorphic functions to detect the boundary, compactness of holomorphic hulls, existence of a psh exhaustion, and a boundary-distance condition. A priori these conditions live in different languages, so none of them directly implies the others. The Levi problem asks whether the analytic abundance of holomorphic functions is exactly the same thing as pseudoconvexity; the equivalence theorem is the point where these separate tests are identified. [quotetheorem:3416] [citeproof:3416] This theorem is the culmination of the classical theory of several complex variables. The four conditions are initially of very different characters: (i) is about the global analytic structure of $\mathcal{O}(\Omega)$; (ii) is a convexity condition in function space; (iii) is a differential-geometric condition on a potential function; and (iv) is an explicit condition on the boundary distance function. Their equivalence is a profound structural result with no analogue in dimension one — in $\mathbb{C}^1$, every domain is a domain of holomorphy (any [open set](/page/Open%20Set) admits holomorphic functions with prescribed singularities), so the Levi problem is trivial. The real content of the theorem is that all three non-trivial conditions converge in several variables to define the same natural class of domains. ## What the Equivalence Buys After the equivalence theorem, pseudoconvexity becomes a practical recognition tool rather than just a boundary condition. To show that a domain has the full function theory of a domain of holomorphy, one may verify a psh exhaustion or a Levi-form condition instead of constructing a single global non-extendable holomorphic function by hand. This is why the theorem reorganises the whole course. Holomorphic convexity, psh exhaustion, boundary distance, and non-extension are no longer separate tests; they are interchangeable languages for the same class of domains. Different examples are easier in different languages, and the rest of the chapter uses that flexibility. The detailed weighted $L^2$ mechanism belongs to the later course on $\bar\partial$ estimates. For the present course, the important point is how to use the result: choose the formulation that makes the domain visible, then transfer the conclusion back to holomorphic functions. ## Convex Domains and the Polydisc What does the equivalence theorem teach us about domains we already understood? Convex domains were shown in Chapter 5 to be pseudoconvex via a direct boundary argument. The equivalence theorem now promotes this: every convex domain is automatically a domain of holomorphy. [quotetheorem:3417] [citeproof:3417] The equivalence theorem is not the only perspective on convex domains. Geometrically, convexity supplies supporting hyperplanes at boundary points, so one expects linear functionals to detect the boundary directly. This gives a useful intuition for why convex domains are rich enough in holomorphic functions; the theorem packages that intuition into the invariant domain-of-holomorphy statement without requiring us to repeat the supporting-hyperplane construction here. [example: The Polydisc] The polydisc $\mathbb{D}^n = \{z \in \mathbb{C}^n : |z_j| < 1 \text{ for all } j\}$ satisfies all four conditions in the equivalence theorem. For condition (iii), the function $\phi(z) = \max_j (-\log(1 - |z_j|^2))$ is psh: each term $-\log(1 - |z_j|^2)$ depends psh-ly on $z_j$ and is constant (hence psh) in all other variables, so the maximum of finitely many psh functions is psh. The sublevel set $\{\phi < c\}$ is contained in $\{|z_j|^2 < 1 - e^{-c}\}$ for every $j$, which is compactly contained in $\mathbb{D}^n$. For condition (i), we use the equivalence theorem itself: the psh exhaustion in condition (iii) makes the polydisc pseudoconvex, and pseudoconvexity is equivalent to being a domain of holomorphy. Pointwise, if $p \in \partial\mathbb{D}^n$ and $|p_j|=1$, then $1/(z_j-p_j)$ is a local blow-up function near $p$; the equivalence theorem upgrades these local obstructions to the global domain-of-holomorphy condition. For condition (iv), the boundary distance satisfies $d(z, \partial\mathbb{D}^n) = \min_j(1 - |z_j|)$, giving \begin{align*} -\log d(z, \partial\mathbb{D}^n) = -\log\min_j(1 - |z_j|) = \max_j(-\log(1 - |z_j|)), \end{align*} a maximum of psh functions, hence psh. [/example] The polydisc illustrates that the equivalence is self-consistent for the simplest domain. But since the polydisc is convex, all four conditions hold for the trivial reason that convex implies pseudoconvex. The genuine non-triviality of the Levi problem — the reason it required 40 years to solve — is illustrated by the following example, where forced extension shows that all four conditions fail simultaneously. [example: The Hartogs Figure Is Not a Domain of Holomorphy] The Hartogs figure in $\mathbb{C}^2$ is the domain \begin{align*} H = \{(z_1,z_2): |z_1|<1,\ |z_2|<1,\ |z_2|>1/2\}\cup \{(z_1,z_2): |z_1|<1/2,\ |z_2|<1\}. \end{align*} This is the standard two-dimensional Hartogs figure. By the [Hartogs extension theorem](/theorems/3401) (Chapter 2), every [holomorphic function](/page/Holomorphic%20Function) on $H$ extends to the full bidisc $\mathbb{D}^2$. So $H$ is not a domain of holomorphy. By the equivalence theorem, $H$ must fail to be pseudoconvex. Geometrically, the missing middle region creates a concave Hartogs-type indentation rather than a compact removable set with a connected-complement hypothesis. The point of the example is therefore forced continuation through the Hartogs geometry, not an application of the compact Hartogs-Bochner removable-singularity theorem. [/example] ## The Reinhardt Domain Example The most instructive non-convex examples of domains of holomorphy are Reinhardt domains. These are domains invariant under coordinate-wise phase rotations, and for them the four equivalent conditions reduce to a single convexity condition on the log-modulus image — making the abstract equivalence concrete and computable. A Reinhardt domain $\Omega \subset \mathbb{C}^n$ is invariant under $(z_1, \dots, z_n) \mapsto (e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n)$ for all $\theta_j \in \mathbb{R}$. Such domains are entirely determined by their image under the log-modulus map $(z_1, \dots, z_n) \mapsto (\log|z_1|, \dots, \log|z_n|)$ (off the coordinate hyperplanes). A Reinhardt domain is logarithmically convex if this log-image is convex as a subset of $\mathbb{R}^n$. For Reinhardt domains, all four conditions in the equivalence theorem are equivalent to logarithmic convexity — this is the content that will be developed fully in Chapter 9. [example: The Domain $\Omega_{a,b} = \{ |z_1|^a |z_2|^b < 1 \}$] Let $a, b > 0$ and consider the Reinhardt domain \begin{align*} \Omega_{a,b} = \{ (z_1, z_2) \in (\mathbb{C}^*)^2 : |z_1|^a |z_2|^b < 1 \}. \end{align*} The log-image of $\Omega_{a,b}$ is $\{(s_1, s_2) \in \mathbb{R}^2 : as_1 + bs_2 < 0\}$, where $s_j = \log|z_j|$. This is a half-plane in $\mathbb{R}^2$, hence convex. By the Reinhardt domain criterion, $\Omega_{a,b}$ is pseudoconvex (and hence all four equivalent conditions hold). For holomorphic convexity, the logarithmic criterion is the right tool. It shows that $\Omega_{a,b}$ is a domain of holomorphy directly from the convex half-plane $as_1+bs_2<0$. When $a/b$ is rational, one can also write explicit Laurent-series barriers using integer exponent vectors normal to the boundary line. For arbitrary real $a,b>0$, such integer normal vectors need not exist, so the structural Reinhardt criterion is the clean statement and the explicit series construction should not be forced. Note that $\Omega_{a,b}$ is generally not convex as a subset of $\mathbb{C}^2 \cong \mathbb{R}^4$. For example, when $a=b=1$, the points $(2,1/4)$ and $(1/4,2)$ both lie in $\Omega_{1,1}$, since their coordinate products have modulus $1/2$. Their midpoint is $(9/8,9/8)$, whose coordinate product has modulus $81/64>1$, so it does not lie in $\Omega_{1,1}$. Thus logarithmic convexity is a genuinely different notion from ordinary convexity in the ambient real vector space. [/example] ## Consequences and Structural Corollaries Several important results follow immediately from the equivalence theorem by combining it with the characterisations established in earlier chapters. The point is that once pseudoconvexity, holomorphic convexity, and the domain-of-holomorphy condition are known to be equivalent, results proved in one language can be transported to the others. This is useful because examples are often easiest to check by psh functions, while applications usually ask for holomorphic functions or cohomology vanishing. [quotetheorem:3418] [citeproof:3418] This consequence turns the equivalence theorem into a practical recognition tool: to prove that a domain has the full function theory of a domain of holomorphy, it is often enough to verify pseudoconvexity. The result also explains why the Levi problem is not merely a boundary regularity question; pseudoconvexity can be formulated intrinsically by psh exhaustion even when the boundary is rough. For domains with smooth boundary, however, Levi's original condition is local and differential: it reads the complex curvature of the boundary through the Levi form. The remaining issue is whether that boundary test is only necessary or whether it actually guarantees the global holomorphic richness described above. [quotetheorem:3392] [citeproof:3392] This theorem is exactly the answer to Levi's original 1910 question: the boundary condition he identified is not merely necessary (as he showed) but also sufficient for a domain with $C^2$ boundary to be a domain of holomorphy. [remark: Dolbeault Cohomology Vanishing] Since every domain of holomorphy is pseudoconvex, and Hörmander's theorem gives solvability of $\bar\partial u = \alpha$ on pseudoconvex domains for every $\bar\partial$-closed $(0,1)$-form $\alpha$, one has $H^{p,q}_{\bar\partial}(\Omega) = 0$ for all $q \geq 1$ on any domain of holomorphy. This vanishing is the analytic key underlying the solvability of both Cousin problems (Chapter 7): Cousin I requires $H^{0,1}_{\bar\partial} = 0$, and Cousin II requires the additional vanishing $H^{1,1}_{\bar\partial} = 0$ together with a topological hypothesis. [/remark] [remark: Biholomorphic Invariance of the Class] All four conditions in the equivalence theorem are biholomorphically invariant. A psh exhaustion on $\Omega$ pulls back under a biholomorphism $\varphi: \Omega \to \Omega'$ to a psh exhaustion on $\Omega'$, because pullbacks of psh functions under holomorphic maps are psh (Chapter 4). The domain-of-holomorphy condition is invariant by definition. Therefore the class of domains of holomorphy is a biholomorphically invariant class, and one may compute any of the four characterising conditions on whichever biholomorphic representative is most convenient. [/remark] The equivalence theorem also raises a natural question: in one variable, every domain is automatically a domain of holomorphy, so the Levi problem is trivial. The Levi problem is a genuine phenomenon of several variables, and it asks which domains in $\mathbb{C}^n$ carry the full richness of [holomorphic function](/page/Holomorphic%20Function) theory. Its solution identifies the answer as pseudoconvex domains. The subsequent question — which complex manifolds (not just domains in $\mathbb{C}^n$) carry this richness? — leads to the theory of Stein manifolds, the subject of Course II. [remark: Oka's Coherence Theorem and Sheaf Theory] Oka's original proof of the Levi problem (for $n = 2$, 1942) worked through the theory of coherent analytic sheaves. Oka's coherence theorem asserts that the sheaf $\mathcal{O}$ of holomorphic functions on a domain in $\mathbb{C}^n$ is coherent, which allows global sections to be patched from local data via a resolution of sheaves. This sheaf-theoretic perspective, developed systematically by Cartan and Grauert in the 1950s, led to Cartan's theorems A and B on Stein manifolds — a sweeping generalisation of the Levi problem equivalences. The condition of being a domain of holomorphy in $\mathbb{C}^n$ becomes, in the manifold setting, the condition of being a Stein manifold, and all four equivalences extend to that setting. [/remark] ## Summary: The Convergence of All Threads The main theorem of this chapter — and the central result of the course — establishes the complete equivalence of four conditions on a proper domain $\Omega \subsetneq \mathbb{C}^n$: 1. $\Omega$ is a domain of holomorphy. 2. $\Omega$ is holomorphically convex. 3. $\Omega$ is pseudoconvex (admits a psh exhaustion). 4. The function $-\log d(\cdot, \partial\Omega)$ is plurisubharmonic on $\Omega$. The chain of implications (1) $\Leftrightarrow$ (2) is the [Cartan–Thullen theorem](/theorems/3385) from Chapter 3. The implication (1) $\Rightarrow$ (4) was established in Chapter 4. The implication (4) $\Rightarrow$ (3) is immediate from definitions. The hard implication (3) $\Rightarrow$ (2) — the [solution of the Levi problem](/theorems/3416) — was proved by Oka (1942) for $n = 2$ and Bremermann–Norguet (1954) in general, with Hörmander's $L^2$ approach (1965) providing the definitive modern proof via weighted $\bar\partial$-estimates. The equivalence says that three entirely different invariants — one analytic (global non-extendability of a [holomorphic function](/page/Holomorphic%20Function)), one algebraic (stability of the hull operation on compact sets), and one differential-geometric (a real-valued differential inequality on an exhaustion) — describe the same class of domains. The hard content is the gap between psh exhaustions and holomorphic functions: a psh function satisfies a differential inequality but has no obvious holomorphic content. That a psh exhaustion forces the domain to carry non-extendable holomorphic functions is Oka's theorem, and it is what makes the subject deep. This equivalence defines the natural class of domains in several complex variables — the domains on which the function theory works in full generality: on which Cousin problems are solvable, on which $\bar\partial$-cohomology vanishes, and on which holomorphic functions can be approximated and constructed with controlled growth. Every subsequent development in the theory takes this as its starting point. Chapter 9 explores the equivalence concretely for Reinhardt domains, where pseudoconvexity reduces to logarithmic convexity and the theory becomes fully explicit. Chapter 10 applies the tools developed throughout the course to prove that the ball and the polydisc in $\mathbb{C}^n$ are biholomorphically inequivalent — a landmark result showing that even within the class of domains of holomorphy, the biholomorphic geometry is rich and non-trivial. --- Reinhardt domains clarify the theory by reducing pseudoconvexity to logarithmic convexity. This chapter develops that symmetric laboratory first; the following chapter then pivots to biholomorphic classification, where most domains no longer have such a simple real-coordinate model. # 9. Reinhardt Domains and the Logarithmic Image Chapter 8 closed the Levi problem by showing that pseudoconvexity (Chapter 5), holomorphic convexity (Chapter 3), the domain-of-holomorphy condition, and the plurisubharmonicity of the log-distance function (Chapter 4) are all equivalent. That equivalence was proved in full generality — but the proof made heavy use of $\bar\partial$-theory (Chapters 6–7) and abstraction, and it can be difficult to see concretely which domains qualify. This chapter develops a class of domains — Reinhardt domains — for which the equivalence reduces to something far more transparent: a geometric condition on a subset of $\mathbb{R}^n$. The key tool is the logarithmic image, a map that encodes the modulus data of a Reinhardt domain in real coordinates, and the main theorem asserts that a Reinhardt domain is pseudoconvex if and only if its logarithmic image is convex in the ordinary sense. This gives a complete, computable classification of domains of holomorphy in the Reinhardt class, illustrated by a family of examples ranging from the polydisc to the Hartogs triangle. ## Reinhardt Domains and the Torus Action A [holomorphic function](/page/Holomorphic%20Function) on a domain in $\mathbb{C}^n$ depends on $z = (z_1, \dots, z_n)$, and the modulus $|z_j|$ of each coordinate is the natural real parameter that governs convergence of [power series](/page/Power%20Series). If a domain is invariant under independent phase rotations of each coordinate, then membership in the domain depends only on the moduli $|z_j|$, and the complex geometry collapses entirely into a real picture. This invariance condition is the definition of a Reinhardt domain. [definition: Reinhardt Domain] A domain $\Omega \subset \mathbb{C}^n$ is a **Reinhardt domain** if for every $(z_1, \dots, z_n) \in \Omega$ and every $\theta = (\theta_1, \dots, \theta_n) \in \mathbb{R}^n$, \begin{align*} (e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n) \in \Omega. \end{align*} That is, $\Omega$ is invariant under the action of the $n$-dimensional torus $\mathbb{T}^n = \{(\zeta_1, \dots, \zeta_n) : |\zeta_j| = 1\}$ acting by componentwise multiplication. [/definition] The condition says that $\Omega$ is a union of orbits under the torus action $(\theta_1, \dots, \theta_n) \cdot z = (e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n)$. Since each orbit through a point $z$ is exactly the set $\{(e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n) : \theta \in \mathbb{R}^n\}$, and this orbit depends only on the vector of moduli $(|z_1|, \dots, |z_n|)$, a Reinhardt domain is completely determined by which modulus vectors $(r_1, \dots, r_n) \in [0, \infty)^n$ it contains. For power series centered at the origin, phase symmetry is not the whole story: convergence at one point forces convergence on the entire coordinatewise polydisc below it. This downward-closure property is the extra condition that separates domains naturally controlled by ordinary power-series estimates from Reinhardt domains that may have holes near the coordinate axes. [definition: Complete Reinhardt Domain] A Reinhardt domain $\Omega \subset \mathbb{C}^n$ is a **complete Reinhardt domain** if for every $(z_1, \dots, z_n) \in \Omega$ and every $(w_1, \dots, w_n)$ with $|w_j| \leq |z_j|$ for all $j$, we have $(w_1, \dots, w_n) \in \Omega$. [/definition] A complete Reinhardt domain is thus one whose modulus image is a downward-closed (in the coordinatewise sense) subset of $[0, \infty)^n$: if a modulus vector $(r_1, \dots, r_n)$ belongs to the image, then so does every $(s_1, \dots, s_n)$ with $s_j \leq r_j$. Concretely, a complete Reinhardt domain contains the origin, and it contains the entire polydisc $\{|w_j| \leq |z_j|\}$ around the origin whenever it contains $z$. [example: Polydisc as a Complete Reinhardt Domain] The polydisc $\mathbb{D}^n = \{z \in \mathbb{C}^n : |z_j| < 1 \text{ for all } j\}$ is a complete Reinhardt domain. It is invariant under the torus action (changing phases does not change moduli), and if $|z_j| < 1$ for all $j$ and $|w_j| \leq |z_j|$, then $|w_j| < 1$ for all $j$, so $(w_1, \dots, w_n)$ also belongs to $\mathbb{D}^n$. The polyannulus $\mathbb{A}^n = \{z \in \mathbb{C}^n : r_j < |z_j| < R_j \text{ for all } j\}$ is a Reinhardt domain but not a complete Reinhardt domain: the point with $|z_j| = (r_j + R_j)/2$ belongs to $\mathbb{A}^n$, but the origin does not. [/example] [example: The Hartogs Triangle] The Hartogs triangle is the Reinhardt domain \begin{align*} T = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| < |z_2| < 1\}. \end{align*} This is a Reinhardt domain: if $(z_1, z_2) \in T$, then $|e^{i\theta_1} z_1| = |z_1| < |z_2| = |e^{i\theta_2} z_2|$, so $(e^{i\theta_1} z_1, e^{i\theta_2} z_2) \in T$. However, $T$ is not a complete Reinhardt domain: the origin $(0, 0)$ does not belong to $T$, since for $(0,0)$ to satisfy the defining inequality we would need $0 < 0 < 1$, which is false. A complete Reinhardt domain must contain the origin (it must be downward-closed under the coordinate moduli), so $T$ fails. The Hartogs triangle therefore does not contain the origin and is not a complete Reinhardt domain. [/example] ## The Logarithmic Image A Reinhardt domain $\Omega$ that intersects $(\mathbb{C}^*)^n = \{z : z_j \neq 0 \text{ for all } j\}$ has a natural real representation via the logarithm of each modulus. The logarithm is the right scale because monomials become linear functions of the numbers $\log|z_j|$, and products of radii become sums. This converts multiplicative convergence inequalities into ordinary convex geometry in $\mathbb{R}^n$. [definition: Logarithmic Image] Let $\Omega \subset \mathbb{C}^n$ be a Reinhardt domain. The **logarithmic image** of $\Omega$ is the set \begin{align*} \operatorname{Log}(\Omega) = \{(\log|z_1|, \dots, \log|z_n|) : z \in \Omega,\, z_j \neq 0 \text{ for all } j\} \subset \mathbb{R}^n. \end{align*} [/definition] The map $\operatorname{Log}: (\mathbb{C}^*)^n \to \mathbb{R}^n$ defined by $z \mapsto (\log|z_1|, \dots, \log|z_n|)$ sends each torus orbit (a set of points sharing the same moduli) to a single point. The logarithmic image is therefore the quotient of $\Omega \cap (\mathbb{C}^*)^n$ by the torus action. For a complete Reinhardt domain, the behaviour at the coordinate hyperplanes $\{z_j = 0\}$ is recorded separately: such a domain always contains the origin and various coordinate discs, but the logarithmic image only records the part where all coordinates are nonzero. [example: Logarithmic Image of the Polydisc] For the polydisc $\mathbb{D}^n = \{|z_j| < 1\}$, the logarithmic image is \begin{align*} \operatorname{Log}(\mathbb{D}^n) = \{(s_1, \dots, s_n) \in \mathbb{R}^n : s_j < 0 \text{ for all } j\} = (-\infty, 0)^n. \end{align*} This is the negative orthant, which is a convex subset of $\mathbb{R}^n$. [/example] [example: Logarithmic Image of the Hartogs Triangle] For the Hartogs triangle $T = \{|z_1| < |z_2| < 1\}$, the logarithmic image is \begin{align*} \operatorname{Log}(T) = \{(s_1, s_2) \in \mathbb{R}^2 : s_1 < s_2 < 0\}. \end{align*} This is the region in $\mathbb{R}^2$ below the line $s_1 = s_2$ and below the axis $s_2 = 0$. Geometrically it is a convex cone — the intersection of two half-planes — hence convex. [/example] The examples suggest the key distinction: after passing to logarithmic coordinates, the domains that behave well for holomorphic function theory have no missing straight-line segments. If the logarithmic image has a concavity, Hartogs-type extension tends to fill it in, so the original domain cannot be the natural boundary for all of its holomorphic functions. The next definition names the convexity condition that prevents this failure. [definition: Logarithmically Convex Reinhardt Domain] A Reinhardt domain $\Omega$ is **logarithmically convex** if $\operatorname{Log}(\Omega \cap (\mathbb{C}^*)^n)$ is a convex subset of $\mathbb{R}^n$. [/definition] Logarithmic convexity is the key condition for Reinhardt domains, playing a role analogous to convexity for ordinary domains. The central theorem of this chapter identifies it with pseudoconvexity. ## The Main Equivalence: Pseudoconvexity and Logarithmic Convexity Before stating the main theorem, it is instructive to see what goes wrong when logarithmic convexity fails. Consider the Reinhardt domain \begin{align*} \Omega_0 = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| < 1,\, |z_2| < 2\} \cup \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| < 2,\, |z_2| < 1\}. \end{align*} This is a connected Reinhardt domain shaped like an L in modulus coordinates. Its logarithmic image (restricted to $(\mathbb{C}^*)^2$) is the L-shaped region $V = \{s_1 < 0,\, s_2 < \log 2\} \cup \{s_1 < \log 2,\, s_2 < 0\}$ in $\mathbb{R}^2$. This set is not convex: the point $p = (-0.5, 0.5)$ belongs to $V$ (since $s_1 = -0.5 < 0$ and $s_2 = 0.5 < \log 2$), and the point $q = (0.5, -0.5)$ belongs to $V$ (since $s_1 = 0.5 < \log 2$ and $s_2 = -0.5 < 0$), but their midpoint $(0, 0)$ satisfies neither $s_1 < 0$ nor $s_2 < 0$, so $(0,0) \notin V$. What does this mean for holomorphic functions on $\Omega_0$? The convex hull of $V$ is not the full square $\{s_1<\log 2,\ s_2<\log 2\}$; it is the half-plane \begin{align*} s_1+s_2<\log 2. \end{align*} Equivalently, the Reinhardt hull has nonzero part described by $|z_1z_2|<2$. The missing diagonal region is the part forced in by logarithmic convexification. Thus the failure of logarithmic convexity predicts holomorphic extension to a larger Reinhardt domain, but not to the entire bidisc of radius $2$. The content of the next theorem is that for Reinhardt domains, the abstract condition of pseudoconvexity — which required the existence of a plurisubharmonic exhaustion, a notion from Chapter 5 — reduces to the entirely real-analytic condition of convexity of the logarithmic image. [quotetheorem:3393] [citeproof:3393] The theorem is important because it replaces a global analytic condition with a concrete convexity test in real coordinates. For Reinhardt domains, pseudoconvexity can therefore be checked by drawing the logarithmic image and asking whether line segments stay inside it. This also explains why nonconvex logarithmic pictures are unstable from the viewpoint of holomorphic functions: the missing real-convex pieces are exactly the regions where extension phenomena are expected. In the rest of the chapter, this real convexity condition becomes the bridge between domains of holomorphy and domains of convergence for Laurent series. ## Laurent Series and Domains of Convergence Given that Reinhardt domains are characterised by their modulus vectors, it is natural to ask: which Reinhardt domains arise as the exact domain where some multi-variable series converges? In one variable, [power series](/page/Power%20Series) converge on discs and [Laurent series](/page/Laurent%20Series) on annuli — both of which are (complete or general) Reinhardt domains in $\mathbb{C}^1$. The question in $n$ variables is whether an analogous characterisation holds, and the answer is yes: the natural domains of convergence are exactly the logarithmically convex Reinhardt domains. Complete Reinhardt domains are the natural domains for multivariable [power series](/page/Power%20Series). In one variable, a holomorphic function on a disc has a Taylor series; in $n$ variables, the same phenomenon holds on complete Reinhardt domains because coordinatewise completeness supplies all smaller polydiscs around the origin. [quotetheorem:3419] [citeproof:3419] The coefficient formula is the multivariable Cauchy coefficient formula: the contour integral runs over an $n$-dimensional torus instead of a circle, and the resulting exponents are nonnegative multi-indices. Thus complete Reinhardt domains behave like the several-variable analogue of discs, while Laurent series belong naturally to Reinhardt domains inside $(\mathbb{C}^*)^n$ where coordinate hyperplanes are excluded. The domain of convergence of a general [Laurent series](/page/Laurent%20Series) has a precise characterisation. The obstruction is to understand how infinitely many monomial inequalities combine; in logarithmic coordinates each monomial contributes a linear constraint, so the natural candidate for the answer is convexity of the logarithmic convergence set. [quotetheorem:3420] [citeproof:3420] The theorem is the precise multi-variable analogue of the classical one-variable fact that [power series](/page/Power%20Series) converge on discs and [Laurent series](/page/Laurent%20Series) on annuli. In $n$ variables the analogue of a disc or annulus is a logarithmically convex Reinhardt domain. The proof explains why coefficient growth becomes convex geometry after taking logarithms; the takeaway here is the dictionary between convergence domains and Reinhardt convexity. ## Classification of Reinhardt Domains of Holomorphy Combining the pseudoconvexity criterion with the Levi problem solution from Chapter 8 gives a complete characterisation of domains of holomorphy in the Reinhardt class. The point is to decide when a Reinhardt domain is already the maximal natural domain for its holomorphic functions, rather than being forced by Hartogs extension into a larger logarithmically convex envelope. [quotetheorem:3394] [citeproof:3394] For Reinhardt domains, the question "is this a domain of holomorphy?" — which in general requires understanding the global [holomorphic function](/page/Holomorphic%20Function) theory — reduces to drawing the logarithmic image and asking whether it is convex. This is a striking simplification. [remark: Holomorphic Functions Witnessing Non-Extendability] When a Reinhardt domain $\Omega$ is logarithmically convex, the holomorphic functions witnessing non-extendability are often explicitly available. On the [domain of convergence of a Laurent series](/theorems/3420), the series itself is a [holomorphic function](/page/Holomorphic%20Function) that cannot be continued across the boundary (since the boundary is precisely where convergence fails). The monomials $z^\alpha$ with $|a_\alpha|$ large for $\alpha$ pointing in a certain direction force blow-up along the corresponding boundary direction in logarithmic coordinates. This makes Reinhardt domains particularly tractable as testing grounds for the general theory. [/remark] ## Examples and the Role of Logarithmic Convexity With the criterion in hand, checking whether a given Reinhardt domain is a domain of holomorphy reduces to a computation in $\mathbb{R}^n$: draw the logarithmic image and determine whether it is convex. The following examples work through this computation for several concrete domains — the bidisc, the ball, the Hartogs triangle, and a domain where the criterion fails — to illustrate both how the check works and what the geometry looks like. [example: The Bidisc and the Ball] The bidisc $\mathbb{D}^2 = \{|z_1| < 1, |z_2| < 1\}$ has logarithmic image $(-\infty, 0)^2$, the open negative quadrant. This is convex, so $\mathbb{D}^2$ is a domain of holomorphy — consistent with the direct verification in Chapter 8. The ball $\mathbb{B}^2 = \{|z_1|^2 + |z_2|^2 < 1\}$ is a Reinhardt domain (it is invariant under the torus action, since rotating phases does not change moduli) — in fact it has a larger symmetry group, the full unitary group $U(2)$. Its intersection with $(\mathbb{C}^*)^2$ has logarithmic image \begin{align*} \operatorname{Log}(\mathbb{B}^2 \cap (\mathbb{C}^*)^2) = \{(s_1, s_2) : e^{2s_1} + e^{2s_2} < 1\}. \end{align*} The boundary of this set in $\mathbb{R}^2$ is the curve $e^{2s_1} + e^{2s_2} = 1$, which is a level set of the convex function $(s_1, s_2) \mapsto e^{2s_1} + e^{2s_2}$ (convex since $e^{2s_j}$ is convex in $s_j$). The sublevel set of a convex function is convex, so $\operatorname{Log}(\mathbb{B}^2 \cap (\mathbb{C}^*)^2)$ is convex. The ball is therefore a domain of holomorphy by the Reinhardt criterion. This conclusion is structural: it does not require writing down a single elementary formula with singularities at every boundary point. [/example] [example: The Hartogs Triangle Is a Domain of Holomorphy] The Hartogs triangle $T = \{|z_1| < |z_2| < 1\}$ has logarithmic image $\{(s_1, s_2) : s_1 < s_2 < 0\}$, which is an open convex subset of $\mathbb{R}^2$ (the intersection of the two half-planes $\{s_1 < s_2\}$ and $\{s_2 < 0\}$). By the criterion, $T$ is a domain of holomorphy. The convexity of the log-image is the reason $T$ is a domain of holomorphy. One can also see explicit boundary obstruction at particular points. On $T$ we have $|z_1/z_2| < 1$ (since $|z_1| < |z_2|$), so the geometric series \begin{align*} g(z_1, z_2) = \sum_{k=0}^\infty \left(\frac{z_1}{z_2}\right)^k = \frac{z_2}{z_2 - z_1} \end{align*} converges absolutely on all of $T$ and defines a [holomorphic function](/page/Holomorphic%20Function) there. It blows up as $z_1/z_2\to 1$, so it obstructs extension across diagonal boundary points with $z_1=z_2\neq 0$. This single function should not be read as detecting every boundary point; the Reinhardt convexity criterion above supplies the global domain-of-holomorphy conclusion. [/example] [example: Two Competing Reinhardt Domains] Consider two Reinhardt domains in $\mathbb{C}^2$: \begin{align*} \Omega_1 &= \{(z_1, z_2) : |z_1| + |z_2| < 1\}, \\ \Omega_2 &= \{(z_1, z_2) : |z_1 z_2| < 1,\, |z_1| < 1,\, |z_2| < 1\}. \end{align*} For $\Omega_1$: the modulus condition $|z_1| + |z_2| < 1$ defines a complete Reinhardt domain (containing the origin). The logarithmic image is \begin{align*} \operatorname{Log}(\Omega_1 \cap (\mathbb{C}^*)^2) = \{(s_1, s_2) : e^{s_1} + e^{s_2} < 1\}. \end{align*} The boundary is $\{e^{s_1} + e^{s_2} = 1\}$. The function \begin{align*} F(s_1,s_2)=e^{s_1}+e^{s_2} \end{align*} is convex, and its sublevel set $\{F<1\}$ is convex: if $F(x)<1$ and $F(y)<1$, then for $0\leq t\leq 1$, \begin{align*} F(tx+(1-t)y)\leq tF(x)+(1-t)F(y)<1. \end{align*} Therefore $\Omega_1$ is logarithmically convex and is a domain of holomorphy by the Reinhardt criterion. For $\Omega_2$: the logarithmic image is \begin{align*} \operatorname{Log}(\Omega_2 \cap (\mathbb{C}^*)^2) = \{(s_1, s_2) : s_1 + s_2 < 0,\, s_1 < 0,\, s_2 < 0\}. \end{align*} This is the intersection of three half-planes in $\mathbb{R}^2$: the negative quadrant $\{s_1 < 0, s_2 < 0\}$ intersected with the half-plane $\{s_1 + s_2 < 0\}$. For $s_1 < 0$ and $s_2 < 0$, the sum $s_1 + s_2 < 0$ holds automatically, so the third constraint is redundant and the log-image is simply $\{s_1 < 0, s_2 < 0\}$ — the same as for $\mathbb{D}^2$. Indeed, if $|z_j| < 1$ for both $j$, then $|z_1 z_2| = |z_1||z_2| < 1$, so the condition $|z_1 z_2| < 1$ is implied by $|z_j| < 1$. Thus $\Omega_2 = \mathbb{D}^2$, which is a domain of holomorphy. A more interesting comparison arises with \begin{align*} \Omega_3 = \{(z_1, z_2) \in (\mathbb{C}^*)^2 : |z_1 z_2| < 1\}. \end{align*} The logarithmic image of $\Omega_3$ is $\{(s_1, s_2) : s_1 + s_2 < 0\}$, a half-plane, which is convex. So $\Omega_3$ is a domain of holomorphy. The function $f(z_1, z_2) = \sum_{k=0}^\infty (z_1 z_2)^k$ converges on $\Omega_3$ (where $|z_1 z_2| < 1$) to the [holomorphic function](/page/Holomorphic%20Function) $(1 - z_1 z_2)^{-1}$, which blows up at the boundary $\{|z_1 z_2| = 1\}$. [/example] [example: A Reinhardt Domain That Is Not a Domain of Holomorphy] Consider the L-shaped Reinhardt domain in $\mathbb{C}^2$ defined by \begin{align*} \Omega_{\text{nc}} = \{|z_1| < 1,\, |z_2| < 4\} \cup \{|z_1| < 4,\, |z_2| < 1\}. \end{align*} The logarithmic image (restricted to $(\mathbb{C}^*)^2$) is the L-shaped region $V = \{s_1 < 0,\, s_2 < \log 4\} \cup \{s_1 < \log 4,\, s_2 < 0\}$. This set is not convex: $p = (-1, 1) \in V$ (via the first part, since $s_1 = -1 < 0$ and $s_2 = 1 < \log 4$) and $q = (1, -1) \in V$ (via the second part, since $s_1 = 1 < \log 4$ and $s_2 = -1 < 0$), but the midpoint $(0, 0)$ satisfies neither $s_1 < 0$ nor $s_2 < 0$, so $(0, 0) \notin V$. By the criterion, $\Omega_{\text{nc}}$ is not a domain of holomorphy. The convex hull of $V$ is the half-plane \begin{align*} s_1+s_2<\log 4, \end{align*} not the full square $\{s_1<\log 4,\ s_2<\log 4\}$. Hence the Reinhardt hull has nonzero part $|z_1z_2|<4$. Holomorphic functions on $\Omega_{\text{nc}}$ extend to this larger logarithmically convex Reinhardt hull; the point is the forced filling of the diagonal gap in logarithmic coordinates, not extension to the whole bidisc of radius $4$. This example illustrates the general principle: when the log-image of a Reinhardt domain is not convex, the domain has a non-trivial "Reinhardt hull" — the Reinhardt domain whose nonzero logarithmic image is the convex hull, with coordinate-hyperplane behaviour handled separately — and holomorphic functions extend to that hull. [/example] ## Pseudoconvex Reinhardt Domains: Complete Structure The preceding sections have established two results: the pseudoconvexity criterion (a Reinhardt domain is a domain of holomorphy iff its log-image is convex) and the domain-of-convergence theorem (the natural domain of a [Laurent series](/page/Laurent%20Series) is always logarithmically convex). It is natural to ask whether these two results are in fact the same statement viewed from two different angles — and the answer is yes. This section makes that identification precise and extracts the complete structure theorem for pseudoconvex Reinhardt domains. [quotetheorem:3421] [citeproof:3421] This theorem closes the circle for Reinhardt domains inside $(\mathbb{C}^*)^n$, where no coordinate hyperplane is present and Laurent monomials are legitimate in every coordinate. For Reinhardt domains in all of $\mathbb{C}^n$, the nonzero part $\Omega \cap (\mathbb{C}^*)^n$ still carries the logarithmic convexity information, but behaviour along coordinate hyperplanes must be treated as extra data rather than hidden inside finite logarithmic coordinates. The domain-of-convergence assertion should also be read in logarithmic coordinates: convergence is controlled by convex inequalities coming from the exponents of the Laurent series. Thus the theorem is not saying that arbitrary domains have Laurent expansions; it says that in the punctured Reinhardt setting, torus symmetry and logarithmic convexity describe exactly the possible domains of convergence. In particular, logarithmic non-convexity is the signal that holomorphic functions are forced to extend to a larger Reinhardt hull. Conversely, logarithmic convexity prevents this particular forced Reinhardt enlargement and is precisely what makes the domain a Reinhardt domain of holomorphy. An important refinement concerns complete Reinhardt domains: on these, [Laurent series](/page/Laurent%20Series) reduce to [power series](/page/Power%20Series) (since the domain contains the origin and coordinate discs), and the domain of convergence of a [power series](/page/Power%20Series) $\sum_{\alpha \in \mathbb{N}^n} a_\alpha z^\alpha$ is always a complete logarithmically convex Reinhardt domain. The exact boundary is governed by growth estimates for the coefficients, not only by the set of exponents that occur. [remark: Comparison with the One-Variable Theory] In one variable, the domains of convergence of [power series](/page/Power%20Series) are discs $\{|z| < R\}$ (complete Reinhardt domains in $\mathbb{C}^1$), and [Laurent series](/page/Laurent%20Series) converge on annuli $\{r < |z| < R\}$ (Reinhardt domains in $\mathbb{C}^1$). The convexity condition on the logarithmic image is automatic in one dimension: a subset of $\mathbb{R}^1$ is convex if and only if it is an interval. So every Reinhardt domain in $\mathbb{C}^1$ is logarithmically convex, and every domain in $\mathbb{C}^1$ is a domain of holomorphy — consistent with the one-variable fact that plane domains support holomorphic functions with boundary singularities. The phenomenon of non-logarithmically-convex Reinhardt domains is genuinely a several-variable phenomenon. [/remark] ## The Holomorphic Hull of a Reinhardt Domain Given a domain of holomorphy $\Omega$ and a compact set $K \Subset \Omega$, the holomorphic hull $\hat{K}_\Omega$ measures how much holomorphic functions on $\Omega$ can "see" the location of $K$. For general domains this hull is an abstract object — one must take the sup-norm over the entire algebra $\mathcal{O}(\Omega)$. For Reinhardt domains, torus symmetry gives a more geometric description, but one has to remember the coordinate hyperplanes: in complete Reinhardt domains, hulls also fill in the coordinatewise polydiscs beneath the modulus data, not only the logarithmic convex hull inside $(\mathbb{C}^*)^n$. Recall that the holomorphic hull of a compact $K$ in a domain $\Omega$ is $\hat{K}_\Omega = \{z \in \Omega : |f(z)| \leq \sup_K |f| \text{ for all } f \in \mathcal{O}(\Omega)\}$. In the punctured Reinhardt setting $\Omega \subseteq (\mathbb{C}^*)^n$, where Laurent monomials are available in positive and negative powers, the clean formula is logarithmic: \begin{align*} \operatorname{Log}(\widehat K_\Omega)=\operatorname{conv}(\operatorname{Log}K)\cap\operatorname{Log}\Omega \end{align*} for torus-invariant compact sets $K \Subset \Omega$. Complete Reinhardt domains require an additional feature: coordinatewise filling toward the origin. Once coordinate hyperplanes are present, the punctured logarithmic formula alone no longer determines the hull, because ordinary holomorphic monomials have only nonnegative powers and the maximum modulus principle fills in lower-modulus points. This distinction explains why monomials are so useful but also why the coordinate axes require care. In a polyannulus, Laurent monomials separate Reinhardt orbits in the same way that linear functionals separate points of a convex set in $\mathbb{R}^n$. In a complete Reinhardt domain such as the bidisc, ordinary holomorphic monomials also force the usual maximum-modulus filling in the missing lower-modulus points. Thus Reinhardt hulls are geometrically transparent, but not quite naive: logarithmic convexification must be combined with the downward filling imposed by holomorphic functions on complete Reinhardt domains. This is why Reinhardt domains are the natural laboratory for holomorphic hull theory. [example: A Torus and Its Holomorphic Hull] Let $n = 2$ and consider the compact torus \begin{align*} K = \{(z_1, z_2) : |z_1| = 1/2,\, |z_2| = 1/2\} \subset \mathbb{D}^2. \end{align*} The logarithmic image of $K$ is the single point $(-\log 2, -\log 2)$, but the holomorphic hull in the bidisc is not just the torus. By the maximum modulus principle in each coordinate, \begin{align*} \widehat{K}_{\mathbb{D}^2} = \{(z_1,z_2) : |z_1| \leq 1/2,\ |z_2| \leq 1/2\}. \end{align*} Indeed, every holomorphic function on the bidisc is bounded on this closed smaller bidisc by its supremum on the distinguished boundary $K$, using the Cauchy integral formula on the polydisc of radius $1/2$. If instead one works in a polyannulus that excludes the coordinate axes, Laurent monomials are available in both positive and negative powers, and the hull can behave more like a logarithmic convex hull without downward filling to the origin. The contrast between the bidisc and a polyannulus is exactly why coordinate hyperplanes have to be treated carefully in Reinhardt hull formulae. [/example] The Reinhardt domain theory presented in this chapter reveals the deep geometric structure behind holomorphy. The logarithmic image map reduces the complex-analytic condition of pseudoconvexity to the real-analytic condition of ordinary convexity in $\mathbb{R}^n$, and the theory of [Laurent series](/page/Laurent%20Series) shows that the domains of holomorphy in the Reinhardt class are exactly the natural domains for multi-variable [power series](/page/Power%20Series). Chapter 10 applies the tools developed throughout the course to prove Poincaré's theorem: the unit ball and the bidisc in $\mathbb{C}^2$ are biholomorphically inequivalent, demonstrating that even among domains of holomorphy the biholomorphic classification is a genuinely harder problem than in one variable. --- The biholomorphic inequivalence of the ball and the bidisc opens the door to deeper questions about the geometry of domains in several complex variables, connecting to Stein manifolds, $L^2$ methods, and curvature theory in subsequent courses. # 10. Biholomorphic Inequivalence and Outlook The [solution of the Levi problem](/theorems/3416) in Chapter 8 established that pseudoconvexity, holomorphic convexity, and the domain-of-holomorphy condition are all equivalent. A natural question follows: given two domains that both satisfy these conditions, must they be biholomorphically equivalent — that is, can a biholomorphic map always be found between them? In one variable, the Riemann mapping theorem answers yes for simply connected proper subdomains of $\mathbb{C}$. In several variables, the answer is emphatically no, and the first counterexample is as fundamental as the theory itself. Poincaré proved in 1907 that the unit ball $B^2 \subset \mathbb{C}^2$ and the bidisc $D^2 \subset \mathbb{C}^2$ — both domains of holomorphy, both pseudoconvex, both simply connected — are not biholomorphically equivalent. This failure of the Riemann mapping theorem shapes the entire subsequent development of the subject. ## The Riemann Mapping Theorem and Its Failure In one variable, every simply connected proper subdomain of $\mathbb{C}$ is biholomorphic to the unit disc $D \subset \mathbb{C}$. The proof uses the fact that the automorphism group of the disc acts transitively and that the uniformisation theory is rigid enough to force equivalence from topological data alone. Several of the ingredients in this argument depend crucially on dimension one. The two most natural simply connected bounded domains in $\mathbb{C}^2$ are the unit ball and the bidisc. [definition: Unit Ball and Bidisc in $\mathbb{C}^2$] The unit ball in $\mathbb{C}^2$ is \begin{align*} B^2 = \{ (z_1, z_2) \in \mathbb{C}^2 : |z_1|^2 + |z_2|^2 < 1 \}. \end{align*} The bidisc in $\mathbb{C}^2$ is \begin{align*} D^2 = \{ (z_1, z_2) \in \mathbb{C}^2 : |z_1| < 1 \text{ and } |z_2| < 1 \}. \end{align*} [/definition] The ball and bidisc are both bounded, simply connected, and pseudoconvex, so one-variable intuition would suggest they might be equivalent models. The obstruction is that several complex variables remembers product structure and boundary geometry, not just simple connectedness. Poincaré's theorem gives the concrete failure of the Riemann mapping theorem in dimension two. [quotetheorem:3396] [citeproof:3396] The theorem is really a statement about the geometry encoded by automorphism groups. The ball has a highly homogeneous geometry — every point looks like every other — while the bidisc has a product structure that creates a preferred coordinate decomposition no biholomorphism can eliminate. One concrete consequence is that no bounded domain in $\mathbb{C}^n$ ($n \geq 2$) serves as a universal model in the way that the unit disc does in one variable. The biholomorphic classification of bounded domains in several variables depends on boundary geometry in ways that have no one-variable counterpart. ## Automorphisms of the Ball What does a biholomorphism of $B^2$ with itself look like, and how many are there? In one variable, automorphisms of the disc form a three-real-parameter family — the Möbius transformations. In two variables, the automorphisms of $B^2$ include an analogous family of Möbius-type maps, and one needs an explicit formula for them to verify the transitivity that underlies Poincaré's proof. The key family generalises the one-variable automorphism $z \mapsto (z - a)/(1 - \bar{a}z)$ of the unit disc. [definition: Ball Automorphisms] For $a = (a_1, a_2) \in B^2$, define $\varphi_a: B^2 \to B^2$ by \begin{align*} \varphi_a(z) = \frac{a - P_a z - s_a Q_a z}{1 - \langle z, a \rangle}, \end{align*} where $s_a = \sqrt{1 - |a|^2}$, $P_a$ is the [orthogonal projection](/theorems/437) onto the span of $a$ in $\mathbb{C}^2$ (so $P_a z = \frac{\langle z, a \rangle}{|a|^2} a$ for $a \neq 0$, and $P_0 = 0$), $Q_a = I - P_a$ is the complementary projection, and $\langle z, a \rangle = z_1 \overline{a_1} + z_2 \overline{a_2}$ is the standard Hermitian inner product. [/definition] The map $\varphi_a$ is the standard involutive ball automorphism exchanging $a$ and $0$. Thus the ball has automorphisms moving any chosen interior point to the origin, a flexibility that the bidisc does not share in the same coordinate-mixing way. [example: Explicit Ball Automorphism] Take $a = (a_1, 0)$ with $0 < a_1 < 1$. Then $P_a z = z_1 a_1 / a_1^2 \cdot (a_1, 0) = (z_1/a_1) \cdot (a_1, 0) = (z_1, 0)$ (projecting onto the first component), $Q_a z = (0, z_2)$, and $s_a = \sqrt{1 - a_1^2}$. The map becomes \begin{align*} \varphi_a(z_1, z_2) = \left( \frac{a_1 - z_1}{1 - \overline{a_1} z_1},\; \frac{-\sqrt{1 - a_1^2}\, z_2}{1 - \overline{a_1} z_1} \right). \end{align*} The first component is exactly the one-variable Möbius automorphism of $D$ sending $a_1$ to $0$. The second component is a scaled version of $z_2$, with the scaling factor $\sqrt{1 - a_1^2}/(1 - \overline{a_1}z_1)$ which is bounded away from zero and infinity on the closed ball. Setting $a_1 = 1/2$, for instance, gives $\varphi_{(1/2,\, 0)}(1/2, 0) = (0, 0)$, confirming that $a = (1/2, 0)$ is sent to the origin. This explicit specialization shows how a ball automorphism can move the first coordinate and simultaneously rescale the transverse coordinate. That mixing is the geometric feature needed later when comparing the ball with the bidisc. [/example] The identity $1 - |w|^2 = (1 - |a|^2)(1 - |z|^2)/|1 - \langle z, a \rangle|^2$ is the key to everything: it expresses that $\varphi_a$ is an isometry for the Bergman metric on $B^2$, and it immediately shows that the family $\{\varphi_a : a \in B^2\}$ together with the unitary group is rich enough to move any point to any other. Making this transitivity precise requires identifying the full automorphism group. [explanation: Why Ball Automorphisms Are Transitive] The full automorphism group of $B^2$ is generated by the maps $\varphi_a$ (one for each $a \in B^2$) and the unitary transformations $z \mapsto Uz$ for $U \in U(2)$. Transitivity follows immediately: given any $p \in B^2$, the map $\varphi_p$ sends $p$ to $0$, and given any other $q \in B^2$, the map $\varphi_q \circ \varphi_p$ sends $p$ to $q$. The isotropy group at $0$ consists exactly of the unitary maps $z \mapsto Uz$, since these are precisely the biholomorphisms of $B^2$ fixing the origin (by a theorem of Cartan: any automorphism of a bounded domain that fixes a point and has derivative the identity at that point is the identity). The group $U(2)$ is compact. This compactness of the isotropy group, combined with the transitivity, means that $B^2$ is a homogeneous space $G/K$ where $G = \operatorname{Aut}(B^2)$ and $K \cong U(2)$. This is the Riemannian symmetric space structure of $B^2$ as the complex hyperbolic space $\mathbb{CH}^2$. [/explanation] ## Automorphisms of the Bidisc and Product Structure The bidisc is also homogeneous: because automorphisms of the unit disc act transitively on $D$, product automorphisms move any point of $D^2$ to any other point. Homogeneity therefore cannot distinguish the bidisc from the ball. The real distinction is that every bidisc automorphism respects the product decomposition, up to swapping the two coordinates, while ball automorphisms mix the coordinates through the larger unitary symmetry. [quotetheorem:3422] [citeproof:3422] The theorem says that the bidisc remembers its two one-dimensional factors. An automorphism may apply an arbitrary disc automorphism in each coordinate and may swap the coordinates, but it cannot blend the variables the way a unitary map blends coordinates in the ball. This rigid product structure is the key contrast with $B^2$. For any $\phi \in \operatorname{Aut}(D^2)$, writing $\phi(z_1, z_2) = (\mu_1(z_1), \mu_2(z_2))$ after possibly swapping coordinates, the two coordinates are transformed independently: \begin{align*} |\phi_1(z_1)| = |\mu_1(z_1)|, \quad |\phi_2(z_2)| = |\mu_2(z_2)|. \end{align*} Thus the isotropy group at the origin is essentially $U(1)^2 \rtimes S_2$, not the full unitary group $U(2)$ that appears for the ball. Equivalently, the boundary of $D^2$ decomposes into edge faces and a corner set, and any automorphism of $D^2$ preserves this stratification up to coordinate permutation. The ball $B^2$ has no corresponding product boundary structure. [example: A Point Where the Orbit Structures Disagree] The product formula also explains what happens at the boundary after taking the standard continuous extensions of disc automorphisms to the closed disc. A point on an edge, such as $(1,w)$ with $|w|<1$, is sent to another edge point because the first coordinate remains on $\partial D$ while the second remains in $D$. A corner point, such as $(1,1)$, is sent to a corner point because both coordinates remain on $\partial D$. Thus edge points and corner points cannot lie in the same boundary orbit, except for the coordinate swap that exchanges the two edge families. This finer stratification of $\partial D^2$ has no analogue for $\partial B^2$, whose boundary has a single orbit under the automorphism group of the ball. [/example] This stratification of the boundary — into corner set, edge components, and their interactions — has no analogue for $B^2$, whose boundary is a smooth strictly pseudoconvex hypersurface with a single orbit. It is not an accident that non-transitivity and boundary stratification coincide: both reflect the same underlying geometry, and the connection between them can be made precise through the theory of CR structures at the boundary. [remark: The Boundary Structure Is the Key Invariant] The deepest way to understand Poincaré's theorem is through the boundary geometry. The boundary $\partial B^2$ is a strongly pseudoconvex hypersurface: the Levi form of its defining function $\rho = |z_1|^2 + |z_2|^2 - 1$ has all eigenvalues equal to $1$ at every boundary point. The boundary $\partial D^2 = \overline{D} \times \partial D \cup \partial D \times \overline{D}$ is a union of two faces, each of which is only weakly pseudoconvex (the defining function for a face such as $\{|z_1| = 1\}$ has a one-dimensional complex tangent space, and the Levi form restricted to this space vanishes). Biholomorphisms preserve the [boundary regularity](/theorems/99) type, so $B^2$ and $D^2$ cannot be biholomorphic. [/remark] ## Proper Holomorphic Maps If no biholomorphism exists between $B^2$ and $D^2$, what is the weakest notion of holomorphic map that can still detect meaningful geometric relationships between domains? Biholomorphisms are too rigid — their non-existence says the domains are inequivalent, but it does not say how different they are. The right generalisation is the class of proper holomorphic maps, which includes finite-to-one branched coverings and captures a coarser notion of equivalence. [definition: Proper Holomorphic Map] A holomorphic map $f: \Omega_1 \to \Omega_2$ between domains $\Omega_1, \Omega_2 \subset \mathbb{C}^n$ is **proper** if for every compact set $K \subset \Omega_2$, the preimage $f^{-1}(K)$ is compact in $\Omega_1$. [/definition] Equivalently, $f$ is proper if whenever a sequence $(z_k) \subset \Omega_1$ converges to $\partial\Omega_1$, the image sequence $f(z_k)$ converges to $\partial\Omega_2$. Proper maps are the right generalisation of biholomorphisms for studying the relationship between domains: a biholomorphism is automatically proper, but proper maps can also be finite-to-one branched coverings. The next theorem asks whether the ball permits any such extra self-maps, or whether its boundary geometry forces every proper self-map to be an automorphism. [quotetheorem:3397] [citeproof:3397] [Alexander's theorem](/theorems/3397) says that the ball admits no hidden finite-sheeted self-coverings. Proper maps may be branched coverings on many domains, but the strong boundary geometry of $B^n$ forces a proper self-map to be one-to-one and onto in the biholomorphic sense. This rigidity is a direct reflection of strong pseudoconvexity and has no analogue for weakly pseudoconvex product domains like the bidisc. [Alexander's theorem](/theorems/3397) is sharp: it fails for the bidisc. There exist proper holomorphic self-maps of $D^2$ that are not automorphisms — for example, $(z_1, z_2) \mapsto (z_1^2, z_2)$ is a proper map $D^2 \to D^2$ (since if $(z_1^2, z_2)$ approaches the boundary of $D^2$ then either $|z_1^2| \to 1$, forcing $|z_1| \to 1$, or $|z_2| \to 1$) but it is not injective (since $(\pm a, b)$ map to the same image). [example: Proper Self-Maps of the Disc versus the Ball] In one variable, the map $h: D \to D$ defined by $h(z) = z^2$ is a proper holomorphic self-map that is not an automorphism: it is $2$-to-$1$ on $D \setminus \{0\}$ and therefore not injective. Properness is immediate: if $|z_k| \to 1$ then $|z_k^2| = |z_k|^2 \to 1$ as well. [Alexander's theorem](/theorems/3397) asserts no analogue of $z^2$ can exist for $B^n$ when $n \geq 2$; every proper self-map of the ball must be injective and hence an automorphism. For the bidisc, the map $\phi: D^2 \to D^2$ given by $\phi(z_1, z_2) = (z_1^2, z_2)$ is a proper holomorphic self-map of $D^2$ that is not an automorphism: properness holds because boundary-approaching sequences in $D^2$ satisfy $|z_1| \to 1$ or $|z_2| \to 1$, and in either case $|\phi_1| = |z_1|^2 \to 1$ or $|\phi_2| = |z_2| \to 1$. But $\phi$ is not injective since $\phi(-a, b) = \phi(a, b)$ for any $a \neq 0$. This contrast underlines the geometric difference between the ball and the bidisc: Alexander's rigidity is a direct consequence of the strict pseudoconvexity of $\partial B^n$, a property that $\partial D^2$ does not have along its edge components. [/example] ## Outlook: The Four Courses This course has developed the foundational theory of holomorphic functions in several variables, centred on the [solution of the Levi problem](/theorems/3416). The tools and phenomena encountered here — pseudoconvexity, the $\bar\partial$-operator, Cousin problems, holomorphic convexity — each point toward more powerful theories developed in subsequent courses. ### Stein Manifolds (Course II) The equivalence theorem establishes that domains of holomorphy in $\mathbb{C}^n$ are exactly the pseudoconvex domains. The natural generalisation asks: what is the correct notion of "domain of holomorphy" for a complex manifold, rather than an open subset of $\mathbb{C}^n$? A complex manifold $X$ is a Stein manifold if it satisfies three conditions: it is holomorphically convex (the holomorphic hull of every compact set is compact), it is holomorphically separable (holomorphic functions separate points), and it admits holomorphic local coordinates (which follows from the other two for complex manifolds). Stein manifolds are precisely the complex manifolds on which the analogy with domains of holomorphy holds: Cousin problems are solvable, the Dolbeault cohomology $H^{p,q}_{\bar\partial}(X) = 0$ for $q \geq 1$, and holomorphic functions can be approximated by globally defined functions on compact subsets. Every domain of holomorphy in $\mathbb{C}^n$ is a Stein manifold, and every Stein manifold embeds biholomorphically into some $\mathbb{C}^N$ (the Remmert embedding theorem). The theory of Stein manifolds also includes Cartan's theorems A and B, which give cohomological conditions for solvability of the $\bar\partial$-equation and of related global problems. ### $L^2$ Methods and Quantitative $\bar\partial$ Theory (Course III) The proof of the Levi problem sketched in Chapter 8 used the solvability of $\bar\partial u = f$ on pseudoconvex domains, appealing to Hörmander's theorem without proof. Course III develops this theory in full, using weighted $L^2$ spaces to give quantitative estimates. The central result is Hörmander's $L^2$ estimate: for a pseudoconvex domain $\Omega \subset \mathbb{C}^n$ and a plurisubharmonic weight function $\phi: \Omega \to \mathbb{R}$, for every $\bar\partial$-closed $(0,1)$-form $f$ with $\int_\Omega |f|^2 e^{-\phi}\, d\mathcal{L}^{2n} < \infty$, there exists a solution $u$ to $\bar\partial u = f$ satisfying \begin{align*} \int_\Omega |u|^2 e^{-\phi}\, d\mathcal{L}^{2n} \leq \int_\Omega |f|^2 e^{-\phi}\, d\mathcal{L}^{2n}. \end{align*} This estimate, proved via the Bochner–Kodaira–Nakano identity, is the engine behind the Levi problem solution and a vast array of other results: the Ohsawa–Takegoshi [extension theorem](/theorems/59), Berndtsson's theorem on positivity of direct images, and the theory of multiplier ideal sheaves. The weight $\phi$ allows analytic results to be read off from geometric data about curvature, setting up the interplay between analysis and geometry in Course IV. ### Curvature and the Neumann Problem (Course IV) The $\bar\partial$-Neumann problem asks for the solution of $\bar\partial u = f$ with minimal $L^2$ norm on a bounded domain, subject to suitable boundary conditions. On a domain with smooth boundary, the analysis of the Neumann operator $N$ (the inverse of $\bar\partial^* \bar\partial + \bar\partial \bar\partial^*$ on $(0,q)$-forms) requires understanding how the complex structure interacts with the boundary geometry. The key condition is again pseudoconvexity, but now understood in terms of curvature: the Levi form of the boundary, which appeared in Chapter 5 as a condition on the complex Hessian of a defining function, reappears as the curvature of the $\overline{CR}$ structure on $\partial\Omega$. The condition of strong pseudoconvexity (all eigenvalues of the Levi form strictly positive) is the correct geometric hypothesis for subelliptic estimates, which give precise regularity for the $\bar\partial$-Neumann problem. The ball $B^2$, being strongly pseudoconvex, is the model domain for this theory. The distinction between $B^2$ and $D^2$ from Poincaré's theorem reappears here with analytic content: $B^2$ is strongly pseudoconvex and the $\bar\partial$-Neumann operator on $B^2$ satisfies subelliptic estimates, while $D^2$ is not strongly pseudoconvex at the edge components of its boundary, and the $\bar\partial$-Neumann problem on $D^2$ requires different methods. The failure of Riemann mapping has consequences for the regularity theory. [remark: The Central Role of Pseudoconvexity] The word "pseudoconvexity" appears in all four courses: as the analytic condition defining domains of holomorphy (Course I), as the geometric condition on Stein manifolds (Course II), as the hypothesis for $L^2$ estimates (Course III), and as the curvature condition for [boundary regularity](/theorems/99) (Course IV). This is not a coincidence. Pseudoconvexity is the complex-geometric analogue of convexity in real analysis: it is the condition that encodes when a domain's geometry is compatible with the structure of holomorphic functions. All of the analysis of the subject flows from it. [/remark] ## Summary and Retrospective This course has traced a complete arc from the basic definition of holomorphic functions in several variables to the [solution of the Levi problem](/theorems/3416), with a final chapter showing that biholomorphic geometry is strictly richer than in one variable. The chain of implications established is: 1. **Hartogs phenomena** (Chapters 1–2): Holomorphic functions in $\mathbb{C}^n$, $n \geq 2$ extend automatically across Hartogs figures and across compact holes when the connected-complement hypotheses of Hartogs--Bochner apply. Not every domain supports non-extendable holomorphic functions. 2. **Domains of holomorphy** (Chapter 3): The correct objects of study are domains on which a non-extendable [holomorphic function](/page/Holomorphic%20Function) exists. The [Cartan–Thullen theorem](/theorems/3385) characterises these as holomorphically convex domains, via the holomorphic hull construction of Chapter 3. 3. **Plurisubharmonic functions** (Chapter 4): The log-distance function $-\log d(\cdot,\partial\Omega)$ is plurisubharmonic on every domain of holomorphy, and the Levi form criterion gives a computable pointwise test. 4. **Pseudoconvexity** (Chapter 5): The existence of a psh exhaustion function gives the geometric condition that the Levi problem will identify with domains of holomorphy; for smooth bounded domains, the Levi form gives a pointwise boundary test for the same condition. 5. **The $\bar\partial$-equation** (Chapter 6) and **Cousin problems** (Chapter 7): The analytic machinery for producing holomorphic functions with prescribed data on domains of holomorphy uses the $\bar\partial$-operator, whose cohomology vanishes on pseudoconvex domains. 6. **The Levi problem** (Chapter 8): The equivalence of domain of holomorphy, holomorphic convexity, pseudoconvexity, and the psh character of the log-distance function is the central theorem. The hard direction — pseudoconvexity implies domain of holomorphy — was proved by Oka and Bremermann–Norguet. 7. **Reinhardt domains** (Chapter 9): For domains symmetric under the torus action, all four equivalent conditions reduce to a single real condition: convexity of the logarithmic image in $\mathbb{R}^n$. 8. **Biholomorphic inequivalence** (this chapter): Even among domains of holomorphy, biholomorphic equivalence is not guaranteed. The ball and the bidisc in $\mathbb{C}^2$ are inequivalent, showing that the classification problem for domains in $\mathbb{C}^n$ is strictly harder than in one variable. The tools developed here — pseudoconvexity, psh functions, $\bar\partial$-cohomology, Cousin problems — form the vocabulary for all of the subsequent theory. The three courses that follow each take one direction in which this vocabulary extends: into geometry (Stein manifolds), into quantitative analysis ($L^2$ estimates), and into the interaction between complex analysis and curvature theory. ## References - L. Hörmander, *An Introduction to Complex Analysis in Several Variables*, 3rd ed., North-Holland, 1990. - R. O. Wells Jr., *Differential Analysis on Complex Manifolds*, 3rd ed., Springer, 2008. - S. G. Krantz, *Function Theory of Several Complex Variables*, 2nd ed., AMS Chelsea, 2001. - K. Fritzsche & H. Grauert, *From Holomorphic Functions to Complex Manifolds*, Springer, 2002.