Several Complex Variables: Foundations¶

Job: Hartogs extension, domains of holomorphy, the $\bar\partial$-equation, and Oka–Cartan coherence — the local-to-global analytic toolkit for $n\ge 2$.

Core¶

Holomorphic functions on $\mathbb{C}^n$¶

$f:U\to\mathbb{C}$ ($U\subset\mathbb{C}^n$ open) is holomorphic if it is holomorphic in each variable separately (then automatically $C^\infty$ and locally given by convergent power series — Osgood / Hartogs theorem on separate analyticity under mild hyp., e.g. local boundedness).

Cauchy integral on polydiscs; identity theorem: agreement on a nonempty open set $\Rightarrow$ agreement on connected components. Zero sets of nonconstant $f$ are thin (complex codimension $1$ analytic sets) — never isolated for $n\ge 2$ unless $f\equiv 0$.

Wirtinger operators $\partial/\partial z_j$, $\partial/\partial\bar z_j$; holomorphy $\Leftrightarrow$ $\bar\partial f=0$.

Hartogs’ extension phenomenon¶

Theorem (Hartogs). Let $n\ge 2$, $\Omega\subset\mathbb{C}^n$ a domain, $K\subset\Omega$ compact with $\Omega\setminus K$ connected. Then every $f\in\mathcal{O}(\Omega\setminus K)$ extends uniquely to $\tilde f\in\mathcal{O}(\Omega)$.

Classical figure. Holomorphic functions on a nbhd of the Hartogs figure in the bidisc extend to the full bidisc. Consequently $\mathcal{O}(\mathbb{C}^n\setminus\{0\})=\mathcal{O}(\mathbb{C}^n)$ for $n\ge 2$ — no isolated singularities.

Removable sets. Analytic sets of codimension $\ge 1$ (with codim $\ge 2$ for bounded holom. functions, Riemann / Hartogs-type removability) are often removable for holomorphic or meromorphic extension under growth control.

Domains of holomorphy¶

$\Omega\subset\mathbb{C}^n$ is a domain of holomorphy if there is no larger domain $\Omega'\supsetneq\Omega$ to which every $f\in\mathcal{O}(\Omega)$ extends. Equivalently (Cartan–Thullen): $\Omega$ is holomorphically convex — for compact $K\subset\Omega$, the holomorphic hull $$ \hat K_\Omega={z\in\Omega:|f(z)|\le\sup_K|f|\ \forall f\in\mathcal{O}(\Omega)} $$ is compact in $\Omega$.

Notion	Role
Holomorphic convexity	$K\mapsto\hat K$ compact in $\Omega$
Pseudoconvexity	local psh defining functions; Levi form $\ge 0$ on analytic tangent space
Levi problem	pseudoconvex $\Leftrightarrow$ domain of holomorphy (solved affirmatively: Oka, solution in full generality mid-20th c.)
Stein manifold	holomorphically convex + holomorphic separation + countable topology; closed complex submanifolds of $\mathbb{C}^N$

Plurisubharmonic (psh) functions: u.s.c. $u$ with $i\partial\bar\partial u\ge 0$ as a current (or $u\circ\gamma$ subharmonic for complex lines $\gamma$). Strictly psh defining functions characterize strictly pseudoconvex boundaries.

The $\bar\partial$-equation¶

For a $(0,q)$-form $u$, solve $\bar\partial v=u$ when $\bar\partial u=0$.

Setting	Result
Polydisc / $\mathbb{C}^n$	Dolbeault lemma: local exactness
Pseudoconvex / Stein	Hörmander $L^2$ estimates: solve $\bar\partial$ with $L^2$ bounds weighted by psh functions
Compact Kähler	Hodge theory replaces pure $L^2$ existence; harmonic representatives

Hörmander. On pseudoconvex $\Omega$, for $\bar\partial$-closed $u\in L^2_{0,q}(\Omega,e^{-\varphi})$ with $\varphi$ strictly psh (weights), there exists $v$ with $\bar\partial v=u$ and $\|v\|^2\le\int |u|^2/(\text{Levi lower bound})$. This is the analytic engine behind Cartan B and vanishing theorems.

Overdetermined system: $\bar\partial^2=0$ is the compatibility condition; ellipticity in the complex-tangential directions ties to microlocal analysis of $\bar\partial_b$ on boundaries.

Oka–Cartan and coherence of $\mathcal{O}$¶

A sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules on a complex space $X$ is coherent if it is locally finitely presented and the kernel of any map $\mathcal{O}_U^p\to\mathcal{F}|_U$ is locally f.g. (standard definition: locally f.g. and relations of finitely many sections are locally f.g.).

Theorem (Oka coherence). The structure sheaf $\mathcal{O}_{\mathbb{C}^n}$ (and $\mathcal{O}_X$ on complex manifolds / reduced complex spaces under standard constructions) is coherent. Ideals of analytic sets are coherent (Rückert Nullstellensatz / Cartan).

Cartan’s Theorems (Stein $X$).

Theorem	Statement
A	For coherent $\mathcal{F}$, stalks $\mathcal{F}_x$ are generated by global sections $\Gamma(X,\mathcal{F})$
B	$H^q(X,\mathcal{F})=0$ for all $q\ge 1$ and all coherent $\mathcal{F}$

Consequences. Surjectivity $\mathcal{O}(X)^p\to\mathcal{O}(X)^q$ criteria; holomorphic convexity characterizations; existence of embeddings of Stein manifolds into $\mathbb{C}^N$; Cousin problems solvable on Stein spaces (additive Cousin $\Leftrightarrow$ $H^1(\mathcal{O})=0$; multiplicative with line bundles).

Oka principle (pointer). On Stein spaces, topological complex vector bundles admit holomorphic structures; many homotopy classes of continuous maps are represented by holomorphic maps (Grauert, Gromov elliptic geometric structures).

See it¶

Cousin problems. Additive Cousin: prescribe principal parts on a cover; obstruction in $H^1(X,\mathcal{O})$. On Stein $X$, Cartan B $\Rightarrow$ solvable. Multiplicative Cousin / Mittag-Leffler for zeros: obstruction in $H^1(X,\mathcal{O}^\times)$, related to Picard / line bundles. For $n=1$, Runge / Weierstrass / Mittag-Leffler are the classical special cases; SCV explains them via coherence + vanishing.

Analytic sets. Zero loci of finitely many holomorphic functions are locally exactly the zero loci of coherent ideal sheaves; dimension theory and Noetherianity of $\mathcal{O}_n$ (stalks are Noetherian local rings — Rückert) parallel algebraic geometry.

Check¶

Show that the punctured polydisc $\Delta^n\setminus\{0\}$ is not a domain of holomorphy for $n\ge 2$.

By Hartogs, every $f\in\mathcal{O}(\Delta^n\setminus\{0\})$ extends to $\Delta^n$, so the holomorphic hull of compact sets surrounding $0$ can escape every compact of the punctured domain; equivalently all functions extend to a strictly larger domain.

Why does Cartan B imply that every holomorphic vector bundle on a Stein manifold is generated by global holomorphic sections?

Apply Theorem A (or B to the sheaf of the bundle and to kernels of surjections from free sheaves): coherence of the sheaf of sections + vanishing of $H^1$ of relation sheaves yields global generation and freeness criteria; finite generation from A, and for the dual: surjections $\mathcal{O}^N\to E$ globally after choosing generators of fibers.

Pitfalls¶

Separate holomorphy without continuity can fail to imply joint holomorphy in pathological cases; local boundedness repairs this.
Pseudoconvexity of the boundary is not the same as holomorphic extendability across a boundary point (local peak functions, etc.).
Coherence fails for $\mathcal{C}^\infty$ structure sheaves in the same algebraic sense — analytic coherence is special to $\mathcal{O}$.
Non-Stein manifolds (e.g. compact positive-dimensional) have abundant coherent sheaf cohomology; Cartan B characterizes Stein among complex manifolds (with separation hyp.).

Refs¶

Hörmander, An Introduction to Complex Analysis in Several Variables
Gunning–Rossi, Analytic Functions of Several Complex Variables
Grauert–Remmert, Theory of Stein Spaces / Coherent Analytic Sheaves
Range, Holomorphic Functions and Integral Representations in Several Complex Variables

Next¶

SCV index · $\bar\partial$-Neumann & Kohn Estimates · Stein & Oka Principle · CR Geometry & $\bar\partial_b$ · Analytic Methods & $L^2$ · Complex geometry foundations · Complex analysis · Commutative algebra · Homological algebra · AG foundations