This course develops the analytic theory of several complex variables through the geometry of real hypersurfaces and CR structures, with a focus on how boundary geometry governs holomorphic behavior. It begins by setting up real hypersurfaces in $\mathbb{C}^n$ and the intrinsic notion of a CR manifold, then uses model spaces such as the Heisenberg group and Siegel domain to illustrate the basic mechanisms behind [boundary regularity](/theorems/99), extension, and invariance. Along the way, the course introduces the tangential [Cauchy-Riemann equations](/page/Cauchy-Riemann%20Equations) as the natural differential system on CR manifolds and shows how they control the local and global behavior of CR functions.

The middle chapters build from local analysis to deeper geometric and PDE phenomena. The Lewy [extension theorem](/theorems/59) and pseudoconvexity at the boundary lead into the Levi problem, explaining when boundary geometry forces holomorphic extendability and domain of holomorphy properties. Chern-Moser theory then provides a normal form and a set of CR invariants for hypersurfaces, while the $\bar{\partial}$-Neumann problem connects these geometric ideas to boundary regularity for the $\bar{\partial}$-operator. The final chapters broaden the perspective by studying CR embeddings, the Boutet de Monvel-Guillemin framework, propagation of CR singularities via the FBI transform, and selected global directions that connect analysis, geometry, and microlocal methods.

# Introduction

This opening chapter fixes the scope of the course and explains why boundary geometry in several complex variables is not a peripheral topic but one of the main places where holomorphic phenomena become visible. In one complex variable, boundaries are often controlled by conformal mapping, harmonic measure, and contour methods; in several variables, real hypersurfaces carry their own complex tangential geometry. The central objects of the course are CR structures, Levi forms, pseudoconvexity, tangential Cauchy-Riemann operators, and the extension or failure of extension of CR functions across boundaries.

The course sits after a first encounter with holomorphic functions on domains in $\mathbb C^n$ and before the more microlocal theory of the $\bar{\partial}$-Neumann problem. We shall repeatedly pass between three languages: geometry of real submanifolds, analysis of differential operators acting only in complex tangential directions, and boundary values of holomorphic functions. The purpose of this introduction is to identify the questions that organise the lectures and to set the notation used throughout.

## The Boundary as a Complex Object

The first problem is to understand what complex analysis can see on a real hypersurface. A real hypersurface $M \subset \mathbb C^n$ has real dimension $2n-1$, so it cannot itself be a complex submanifold. Nevertheless, at each point there are tangent directions that are stable under multiplication by $i$, and those directions form the complex tangent bundle.

[definition: Real Hypersurface] Let $U \subset \mathbb C^n$ be open. A real hypersurface in $U$ is a smooth embedded submanifold $M \subset U$ of real codimension $1$. [/definition]

A hypersurface is therefore too large to be complex and too small to be open. To make the surviving complex directions usable in analysis, we need a pointwise definition of the tangent directions that remain stable under the ambient complex structure.

[definition: Complex Tangent Space] Let $M \subset \mathbb C^n$ be a real hypersurface and let $p \in M$. The complex tangent space at $p$ is \begin{align*} H_pM = T_pM \cap iT_pM. \end{align*} The holomorphic tangent space is the $(1,0)$ part \begin{align*} T_p^{1,0}M = \{L \in T_p^{1,0}\mathbb C^n : L\rho = 0\}, \end{align*} where $\rho$ is any local defining function for $M$ near $p$. [/definition]

The definition refers to a local defining function, but the resulting space does not depend on the chosen defining function. The lectures begin by checking this invariance and then using it to define geometric quantities intrinsic to the hypersurface.

[example: The Unit Sphere] Let $M=\{z\in\mathbb C^n:|z|^2=1\}$ and choose the defining function \begin{align*} \rho(z)=|z|^2-1=\sum_{k=1}^n z_k\bar z_k-1. \end{align*} Fix $p\in M$, and write an arbitrary $(1,0)$ vector at $p$ as \begin{align*} L=\sum_{j=1}^n a_j\partial_{z_j}. \end{align*} For each pair $j,k$, the complex coordinates $z_k$ and $\bar z_k$ are treated as independent when differentiating with respect to $z_j$, so \begin{align*} \partial_{z_j}(z_k\bar z_k)=(\partial_{z_j}z_k)\bar z_k+z_k(\partial_{z_j}\bar z_k)=\delta_{jk}\bar z_k+z_k\cdot 0=\delta_{jk}\bar z_k. \end{align*} Therefore \begin{align*} \partial_{z_j}\rho(z)=\partial_{z_j}\left(\sum_{k=1}^n z_k\bar z_k-1\right)=\sum_{k=1}^n\delta_{jk}\bar z_k-0=\bar z_j. \end{align*} Evaluating at $p$ gives \begin{align*} \partial_{z_j}\rho(p)=\bar p_j. \end{align*} Applying $L$ to $\rho$ at $p$ gives \begin{align*} L\rho(p)=\left(\sum_{j=1}^n a_j\partial_{z_j}\right)\rho(p)=\sum_{j=1}^n a_j\partial_{z_j}\rho(p). \end{align*} Substituting $\partial_{z_j}\rho(p)=\bar p_j$ yields \begin{align*} L\rho(p)=\sum_{j=1}^n a_j\bar p_j. \end{align*} By the definition of the holomorphic tangent space, $L\in T_p^{1,0}M$ exactly when $L\rho(p)=0$, and hence \begin{align*} T_p^{1,0}M=\left\{\sum_{j=1}^n a_j\partial_{z_j}:\sum_{j=1}^n a_j\bar p_j=0\right\}. \end{align*} The equation $\sum_{j=1}^n a_j\bar p_j=0$ is nontrivial. Indeed, since $p\in M$, \begin{align*} \sum_{j=1}^n |p_j|^2=|p|^2=1. \end{align*} If every coefficient $\bar p_j$ were $0$, then every $p_j$ would be $0$, so the left side would be $0$, contradicting $\sum_{j=1}^n |p_j|^2=1$. Thus at least one $\bar p_j$ is nonzero. The map \begin{align*} \sum_{j=1}^n a_j\partial_{z_j}\longmapsto \sum_{j=1}^n a_j\bar p_j \end{align*} is therefore a nonzero complex linear functional on $T_p^{1,0}\mathbb C^n$, and $T_p^{1,0}M$ is its kernel, a complex hyperplane. Next compute the Levi form. Since $\partial_{z_j}\rho=\bar z_j$, differentiating with respect to $\bar z_k$ gives \begin{align*} \frac{\partial^2\rho}{\partial z_j\partial\bar z_k}(p)=\frac{\partial}{\partial\bar z_k}(\bar z_j)(p)=\delta_{jk}. \end{align*} For $L=\sum_{j=1}^n a_j\partial_{z_j}\in T_p^{1,0}M$, the Levi form is \begin{align*} \mathcal L_{\rho,p}(L,\overline L)=\sum_{j,k=1}^n \frac{\partial^2\rho}{\partial z_j\partial\bar z_k}(p)a_j\overline{a_k}=\sum_{j,k=1}^n\delta_{jk}a_j\overline{a_k}. \end{align*} When $j\ne k$, $\delta_{jk}=0$, and when $j=k$, $\delta_{jj}=1$. Hence \begin{align*} \sum_{j,k=1}^n\delta_{jk}a_j\overline{a_k}=\sum_{j=1}^n a_j\overline{a_j}=\sum_{j=1}^n |a_j|^2. \end{align*} Thus \begin{align*} \mathcal L_{\rho,p}(L,\overline L)=\sum_{j=1}^n |a_j|^2. \end{align*} If $L\ne 0$, then some coefficient $a_j$ is nonzero, so the corresponding summand $|a_j|^2$ is positive and every other summand is nonnegative. Therefore $\mathcal L_{\rho,p}(L,\overline L)>0$ for every nonzero $L\in T_p^{1,0}M$, so the unit sphere is the basic model of a strongly pseudoconvex boundary. [/example]

This example already shows the pattern of the course. The ambient holomorphic coordinates give concrete formulas, but the important information is invariant under biholomorphic changes of coordinates.

## Levi Geometry and Pseudoconvexity

The next problem is to measure how the complex tangent directions bend inside the hypersurface. Ordinary curvature is not the right invariant for holomorphic extension questions. The relevant object is the Levi form, a Hermitian form on complex tangent vectors obtained by differentiating a defining function twice in complex directions.

[definition: Levi Form of a Hypersurface] Let $M = \{\rho = 0\} \subset \mathbb C^n$ be a smooth real hypersurface with $d\rho \ne 0$ on $M$. For $p \in M$, the Levi form of $M$ with respect to $\rho$ is the Hermitian form on $T_p^{1,0}M$ given by \begin{align*} \mathcal L_{\rho,p}(L,\overline{K}) = \sum_{j,k=1}^n \frac{\partial^2\rho}{\partial z_j\partial \bar z_k}(p)L_j\overline{K_k}. \end{align*} [/definition]

Changing the defining function rescales the Levi form by a positive factor when the orientation is fixed. Thus the sign of the Levi form is the invariant datum, while its numerical normalization is not. The boundary condition needed for extension theory is precisely the absence of negative complex-tangential curvature on the side from which holomorphic functions are to extend.

[definition: Pseudoconvex Hypersurface] Let $M \subset \mathbb C^n$ be an oriented real hypersurface with a local defining function $\rho$ chosen so that the pseudoconvex side is $\{\rho < 0\}$. The hypersurface is pseudoconvex at $p \in M$ if $\mathcal L_{\rho,p}$ is positive semidefinite on $T_p^{1,0}M$. [/definition]

Pseudoconvexity is the boundary version of holomorphic convexity. In later chapters it will control both local extension results and estimates for the tangential Cauchy-Riemann complex.

[example: Sphere and Hyperquadric] For the sphere $M_S=\{|z|^2=1\}\subset\mathbb C^n$, take \begin{align*} \rho_S(z)=|z|^2-1=\sum_{m=1}^n z_m\bar z_m-1. \end{align*} We compute its Levi form by differentiating $\rho_S$ in the holomorphic and antiholomorphic coordinate directions. Treating $z_m$ and $\bar z_m$ as independent coordinates, for each pair $j,m$, \begin{align*} \partial_{z_j}(z_m\bar z_m)=(\partial_{z_j}z_m)\bar z_m+z_m(\partial_{z_j}\bar z_m)=\delta_{jm}\bar z_m+z_m\cdot 0=\delta_{jm}\bar z_m. \end{align*} Therefore \begin{align*} \frac{\partial\rho_S}{\partial z_j}=\partial_{z_j}\left(\sum_{m=1}^n z_m\bar z_m-1\right)=\sum_{m=1}^n\delta_{jm}\bar z_m-0=\bar z_j. \end{align*} Differentiating with respect to $\bar z_k$ gives \begin{align*} \frac{\partial^2\rho_S}{\partial z_j\partial\bar z_k}=\frac{\partial}{\partial\bar z_k}(\bar z_j)=\delta_{jk}. \end{align*} Let $p\in M_S$ and let \begin{align*} L=\sum_{j=1}^n a_j\partial_{z_j}\in T_p^{1,0}M_S. \end{align*} By the definition of the Levi form, \begin{align*} \mathcal L_{\rho_S,p}(L,\overline L)=\sum_{j,k=1}^n \frac{\partial^2\rho_S}{\partial z_j\partial\bar z_k}(p)a_j\overline{a_k}. \end{align*} Substituting $\frac{\partial^2\rho_S}{\partial z_j\partial\bar z_k}(p)=\delta_{jk}$ yields \begin{align*} \mathcal L_{\rho_S,p}(L,\overline L)=\sum_{j,k=1}^n \delta_{jk}a_j\overline{a_k}. \end{align*} The summand is $0$ when $j\ne k$, and when $j=k$ it is $a_j\overline{a_j}=|a_j|^2$, so \begin{align*} \mathcal L_{\rho_S,p}(L,\overline L)=\sum_{j=1}^n |a_j|^2. \end{align*} If $L\ne 0$, then some coefficient $a_j$ is nonzero, so the corresponding term $|a_j|^2$ is positive and all other terms are nonnegative. Hence the Levi form of the sphere is positive definite on every complex tangent space. For the hyperquadric \begin{align*} M_Q=\{(z_1,z_2,w)\in\mathbb C^3:\operatorname{Im}w=|z_1|^2-|z_2|^2\}, \end{align*} use the defining function \begin{align*} \rho_Q(z_1,z_2,w)=|z_1|^2-|z_2|^2-\operatorname{Im}w=z_1\bar z_1-z_2\bar z_2-\frac{w-\bar w}{2i}. \end{align*} The holomorphic first derivatives are \begin{align*} \frac{\partial\rho_Q}{\partial z_1}=\partial_{z_1}(z_1\bar z_1)-\partial_{z_1}(z_2\bar z_2)-\partial_{z_1}\left(\frac{w-\bar w}{2i}\right)=\bar z_1-0-0=\bar z_1. \end{align*} \begin{align*} \frac{\partial\rho_Q}{\partial z_2}=\partial_{z_2}(z_1\bar z_1)-\partial_{z_2}(z_2\bar z_2)-\partial_{z_2}\left(\frac{w-\bar w}{2i}\right)=0-\bar z_2-0=-\bar z_2. \end{align*} \begin{align*} \frac{\partial\rho_Q}{\partial w}=\partial_w(z_1\bar z_1)-\partial_w(z_2\bar z_2)-\partial_w\left(\frac{w-\bar w}{2i}\right)=0-0-\frac{1}{2i}=\frac{i}{2}. \end{align*} Differentiating these with respect to the antiholomorphic variables gives \begin{align*} \frac{\partial^2\rho_Q}{\partial z_1\partial\bar z_1}=\frac{\partial}{\partial\bar z_1}(\bar z_1)=1. \end{align*} \begin{align*} \frac{\partial^2\rho_Q}{\partial z_2\partial\bar z_2}=\frac{\partial}{\partial\bar z_2}(-\bar z_2)=-1. \end{align*} \begin{align*} \frac{\partial^2\rho_Q}{\partial w\partial\bar w}=\frac{\partial}{\partial\bar w}\left(\frac{i}{2}\right)=0. \end{align*} The remaining mixed second derivatives with distinct variables are $0$: $\bar z_1$ depends only on $\bar z_1$, $-\bar z_2$ depends only on $\bar z_2$, and $i/2$ is constant. Thus for \begin{align*} L=a_1\partial_{z_1}+a_2\partial_{z_2}+b\partial_w\in T_p^{1,0}M_Q, \end{align*} the Levi form is \begin{align*} \mathcal L_{\rho_Q,p}(L,\overline L)=1\cdot a_1\overline{a_1}+(-1)\cdot a_2\overline{a_2}+0\cdot b\overline b=|a_1|^2-|a_2|^2. \end{align*} The positive and negative values occur on actual complex tangent vectors. At $p=(p_1,p_2,p_w)\in M_Q$, the first derivatives above give \begin{align*} \frac{\partial\rho_Q}{\partial z_1}(p)=\bar p_1. \end{align*} \begin{align*} \frac{\partial\rho_Q}{\partial z_2}(p)=-\bar p_2. \end{align*} \begin{align*} \frac{\partial\rho_Q}{\partial w}(p)=\frac{i}{2}. \end{align*} Set \begin{align*} L_+=\partial_{z_1}+2i\bar p_1\partial_w. \end{align*} Then \begin{align*} L_+\rho_Q(p)=1\cdot\bar p_1+(2i\bar p_1)\cdot\frac{i}{2}. \end{align*} Since $(2i)(i/2)=i^2=-1$, \begin{align*} L_+\rho_Q(p)=\bar p_1-\bar p_1=0. \end{align*} Hence $L_+\in T_p^{1,0}M_Q$. In the formula $\mathcal L_{\rho_Q,p}(L,\overline L)=|a_1|^2-|a_2|^2$, this vector has $a_1=1$ and $a_2=0$, so \begin{align*} \mathcal L_{\rho_Q,p}(L_+,\overline{L_+})=|1|^2-|0|^2=1. \end{align*} Similarly, set \begin{align*} L_-=\partial_{z_2}-2i\bar p_2\partial_w. \end{align*} Then \begin{align*} L_-\rho_Q(p)=1\cdot(-\bar p_2)+(-2i\bar p_2)\cdot\frac{i}{2}. \end{align*} Since $(-2i)(i/2)=-i^2=1$, \begin{align*} L_-\rho_Q(p)=-\bar p_2+\bar p_2=0. \end{align*} Hence $L_-\in T_p^{1,0}M_Q$. In the same Levi-form formula this vector has $a_1=0$ and $a_2=1$, so \begin{align*} \mathcal L_{\rho_Q,p}(L_-,\overline{L_-})=|0|^2-|1|^2=-1. \end{align*} The sphere is the model with strictly positive Levi geometry, while this hyperquadric is the basic model where the Levi form has both positive and negative complex tangential directions. [/example]