This course develops the central analytical and topological structures that arise on complex manifolds equipped with Kähler metrics. It begins with the basic geometry of Kähler manifolds and concrete examples, then moves to the differential operators naturally associated to the complex structure, the Lefschetz package from linear algebra, and the identities that tie these pieces together. From there, the course turns to harmonic forms and [Hodge decomposition](/theorems/2745), showing how analytic methods produce powerful cohomological consequences.

A recurring theme is that the Kähler condition imposes remarkable rigidity: the $\partial\bar{\partial}$ lemma, [Hard Lefschetz theorem](/theorems/3876), and Hodge structures all emerge as manifestations of the same underlying symmetry. Later chapters use these results to study low-dimensional geometry, especially Kähler surfaces, and to connect the theory with periods, deformations, and applications. The chapters are arranged so that each new layer builds on the previous one, starting from definitions and examples, then establishing the key operator identities, and finally deriving the structural theorems that make Kähler geometry such a rich bridge between analysis, algebra, and topology.

# Introduction

This course studies the part of complex geometry where a Hermitian metric, a symplectic form, and elliptic analysis reinforce each other. The basic objects are compact Kähler manifolds, and the basic question is how the differential geometry of a metric controls the topology and complex structure of the underlying manifold. The main results of the course are the [Kähler identities](/theorems/3853), Hodge decomposition, Lefschetz theory, and the Hodge-Riemann bilinear relations. Together they explain why compact Kähler manifolds behave much more rigidly than general compact complex manifolds.

The course assumes the language of complex manifolds, vector bundles, differential forms, de Rham cohomology, Hermitian metrics, and Dolbeault cohomology. It also uses the standard analytic fact that elliptic [self-adjoint operators](/page/Self-Adjoint%20Operators) on compact manifolds have finite-dimensional spaces of harmonic forms and good regularity theory. We will recall the geometric meaning of each analytic object when it enters, but the course is not a first course in elliptic PDE. The emphasis is on how the Kähler condition aligns operators that, on a general complex manifold, have no reason to communicate.

## The Central Problem

What extra structure does a closed Hermitian form impose on a complex manifold? A Hermitian metric on a complex manifold always gives a real $(1,1)$-form, but the requirement that this form be closed is a strong compatibility condition. It says that the metric geometry, the complex structure, and the [exterior derivative](/theorems/1525) are fitted together tightly enough that the de Rham and Dolbeault theories become two views of the same object.

The first task is to fix the language used throughout the course. The following definition is the hinge between complex geometry and symplectic geometry.

[definition: Kähler Manifold] A Kähler manifold is a complex manifold $X$ equipped with a Hermitian metric $g$ whose associated real $(1,1)$-form $\omega$ satisfies $d\omega = 0$. [/definition]

After this definition, two viewpoints will be used at the same time. From the complex-geometric viewpoint, $\omega$ is the fundamental form of a Hermitian metric. From the symplectic viewpoint, $\omega$ is a symplectic form compatible with the complex structure. Chapter 1 makes this equivalence between Hermitian metrics and closed positive $(1,1)$-forms explicit, and the rest of the course is built around its consequences.

[example: Projective Space As A Model] On the affine chart $U_0=\{[Z_0:\cdots:Z_n]\mid Z_0\ne 0\}$ with coordinates $z_j=Z_j/Z_0$, set $\rho=1+|z|^2=1+\sum_{j=1}^n z_j\bar z_j$ and $\varphi=\log \rho$. We compute the local Fubini-Study form $\theta_{\mathrm{FS}}=i\partial\bar{\partial}\varphi$ explicitly. Since $\bar{\partial}\rho=\sum_{j=1}^n z_j\,d\bar z_j$, the chain rule gives \begin{align*} \bar{\partial}\varphi=\frac{1}{\rho}\bar{\partial}\rho=\sum_{j=1}^n \frac{z_j}{\rho}\,d\bar z_j. \end{align*} For each $j$, \begin{align*} \partial\left(\frac{z_j}{\rho}\right)=\sum_{k=1}^n \frac{\rho\delta_{jk}-z_j\bar z_k}{\rho^2}\,dz_k. \end{align*} Therefore \begin{align*} \theta_{\mathrm{FS}}=i\sum_{j,k=1}^n \frac{\rho\delta_{jk}-z_j\bar z_k}{\rho^2}\,dz_k\wedge d\bar z_j. \end{align*} For a tangent vector $v=(v_1,\ldots,v_n)$, the associated Hermitian quadratic form is \begin{align*} \sum_{j,k=1}^n \frac{\rho\delta_{jk}-z_j\bar z_k}{\rho^2}v_k\bar v_j=\frac{\rho\sum_{k=1}^n |v_k|^2-\left|\sum_{k=1}^n \bar z_k v_k\right|^2}{\rho^2}. \end{align*} By Cauchy-Schwarz, $\left|\sum_k \bar z_k v_k\right|^2\le |z|^2|v|^2$, so the numerator is at least $(1+|z|^2)|v|^2-|z|^2|v|^2=|v|^2$, and it is positive when $v\ne 0$. The form is closed because \begin{align*} d\theta_{\mathrm{FS}}=i(\partial+\bar{\partial})\partial\bar{\partial}\varphi=0, \end{align*} using $\partial^2=0$, $\bar{\partial}^2=0$, and $\partial\bar{\partial}+\bar{\partial}\partial=0$. On an overlap of two projective charts, changing homogeneous coordinates changes the potential by $\log |f|^2$ for a nowhere-vanishing [holomorphic function](/page/Holomorphic%20Function) $f$, and $i\partial\bar{\partial}\log |f|^2=0$ locally because $\log |f|^2=\log f+\log \bar f$ after choosing a holomorphic branch of $\log f$. Thus these local formulas glue to a global positive closed real $(1,1)$-form on $\mathbb{CP}^n$, showing that the Kähler condition appears naturally in projective geometry. [/example]

The example also indicates why local potentials will matter. A Kähler form is locally controlled by a single real-valued function, so metric data can often be converted into complex differential operators. This is the first place where the course begins to replace coordinate computations by structural identities.

## Why Kähler Geometry Has More Cohomology Than Expected

Why should a metric condition change the topology of a compact complex manifold? The answer comes from [harmonic representatives](/theorems/2747). On any compact Riemannian manifold, Hodge theory identifies de Rham cohomology classes with harmonic forms. On a compact Kähler manifold, the harmonic condition respects the $(p,q)$-decomposition of complex differential forms, so cohomology splits according to complex type.

To state the target structure, we recall the cohomology groups that the course relates. These are not new objects, but their interaction is the point.

[definition: Dolbeault Cohomology] Let $X$ be a complex manifold. The Dolbeault cohomology group $H^{p,q}_{\bar{\partial}}(X)$ is the cohomology of the complex \begin{align*} \Omega^{p,q}(X) \xrightarrow{\bar{\partial}} \Omega^{p,q+1}(X) \xrightarrow{\bar{\partial}} \Omega^{p,q+2}(X). \end{align*} [/definition]

In these notes, $\Omega^{p,q}(X)$ and $A^{p,q}(X)$ both denote the smooth complex-valued $(p,q)$-forms on $X$; later chapters mostly use the notation $A^{p,q}(X)$ when discussing differential operators.

Dolbeault cohomology records the failure of $(p,q)$-forms to be solved by the $\bar{\partial}$-operator. In a general complex manifold it is related to de Rham cohomology through spectral sequences and filtrations. In the Kähler case, those hidden filtrations split into an actual direct sum, which is the first major reason compact Kähler manifolds have strong topological restrictions.

The compactness hypothesis is part of the analytic mechanism: without compactness, harmonic representatives need not give finite-dimensional cohomology or a clean global decomposition. The Kähler hypothesis is also essential, since compact complex manifolds such as Hopf surfaces have $b_1=1$, contradicting the evenness forced by the degree-one decomposition below. The decomposition does not say that every complex manifold has cohomology split by type, nor does it identify the ring structure on de Rham cohomology; it is an additive decomposition of complex vector spaces. For now we use it as the organizing target of the course; when the [Hodge decomposition theorem](/theorems/3941) is invoked later, subsequent appearances will refer back to that result rather than restating the full theorem each time.

The first obstruction worth isolating is degree $1$. This degree-one part of the decomposition is not only a structural description of cohomology groups; together with complex conjugation, it imposes an arithmetic constraint on the ordinary Betti number $b_1(X)$. Naming this consequence separately is useful because it gives a fast test for non-Kähler compact complex manifolds, before any of the later Lefschetz machinery is available.