Several Complex Variables

Job: holomorphic functions of \(n\ge 2\) complex variables — Hartogs extension, domains of holomorphy, \(\bar\partial\), and the Oka–Cartan coherence package.

Core

For \(n\ge 2\), holomorphy is nonlocal in a way impossible in one variable: compact holes can be filled in (Hartogs), and the natural domains of existence are pseudoconvex / holomorphically convex, not arbitrary opens. Coherent analytic sheaves and Cartan’s Theorems A/B organize the global theory.

Layer Content
Local holomorphy power series, CR, identity theorem in \(\mathbb{C}^n\)
Extension Hartogs; removable singularities
Domains domains of holomorphy, holomorphic convexity, Levi / pseudoconvexity
\(\bar\partial\) Dolbeault, solvability, Hörmander \(L^2\)
Oka–Cartan coherence of \(\mathcal{O}\), Theorems A/B on Stein spaces

Reading map

Page Focus
Foundations Hartogs, domains of holomorphy, \(\bar\partial\), Oka–Cartan, coherence of \(\mathcal{O}\)
\(\bar\partial\)-Neumann & Kohn Estimates \(\bar\partial\)-Neumann operator; Kohn subelliptic; Catlin finite type; compactness / regularity; Bergman / Szegő
Stein & Oka Principle Stein characterization / embedding; Grauert–Forster Oka; Gromov ellipticity / spray; Oka vs hyperbolic
CR Geometry & \(\bar\partial_b\) CR / Levi form; \(\bar\partial_b\) / Kohn–Rossi; Lewy extension / non-solvability; Boutet de Monvel embedding; Heisenberg
Analytic Methods & \(L^2\) Hörmander \(L^2\) / Berndtsson; Bochner–Martinelli / Henkin–Ramirez; Bergman asymptotics; Ohsawa–Takegoshi / Nadel

Dependency order

  1. One complex variable — complex analysis.
  2. SCV foundations (this track).
  3. Complex geometry for compact manifolds / Hodge.
  4. Analytic spaces \(\leftrightarrow\) algebraic geometry via GAGA on projective schemes.

Interfaces

Area Contact
Complex analysis \(n=1\); residues do not globalize the same way
Complex geometry manifolds, Dolbeault, Kähler
Commutative algebra stalks of \(\mathcal{O}\), coherence, depth
PDE / microlocal \(\bar\partial\) as overdetermined elliptic system
Functional analysis Fréchet spaces of holomorphic functions, \(L^2\) estimates

Notation (paper-standard)

  • \(\mathbb{C}^n\), polydiscs \(\Delta^n\), \(H(U)=\mathcal{O}(U)\)
  • \(\bar\partial\), \(L^2_{\mathrm{loc}}\), plurisubharmonic (psh), strictly pseudoconvex
  • Stein space, \(\operatorname{Coh}(\mathcal{O}_X)\), Cartan A/B

See it

Hartogs figure. On \(\mathbb{C}^2\), holomorphic functions on a connected nbhd of \((\overline{\Delta}\times\partial\Delta)\cup(\partial\Delta\times\overline{\Delta})\) extend holomorphically to the full bidisc — there is no holomorphic function with singularity set a compact \(K\subset U\) of codimension \(\ge 1\) in the way poles work for \(n=1\). Singularities of holomorphic functions for \(n\ge 2\) are never isolated.

Check

Why can there be no holomorphic function on \(\mathbb{C}^2\setminus\{0\}\) that does not extend through \(0\)?

Hartogs’ extension theorem: for \(n\ge 2\), holomorphic functions on a domain minus a compact subset with connected complement (more precisely: remove a compact \(K\) from a domain \(\Omega\) under standard topological hyp.) extend across \(K\). In particular \(\mathcal{O}(\mathbb{C}^n\setminus\{0\})=\mathcal{O}(\mathbb{C}^n)\) for \(n\ge 2\).

What is the one-variable obstruction that Hartogs removes?

In \(\mathbb{C}\), \(\{0\}\) is a codimension-\(1\) analytic set; Laurent principal parts create genuine non-extendable singularities. For \(n\ge 2\), \(\{0\}\) has codimension \(n\ge 2\), and the \(\bar\partial\) / integral extension machinery fills holes.

Pitfalls

  • Holomorphic convexity \(\neq\) geometric convexity — pseudoconvex domains generalize Euclidean convexity in the complex category.
  • Stein \(\neq\) affine algebraic — every Stein manifold is holomorphically convex and separable by holomorphic functions; affine varieties are Stein after analytification, but not conversely in the non-algebraic world.
  • Coherence is nontrivial: \(\mathcal{O}\) is coherent (Oka); arbitrary sheaves of \(\mathcal{O}\)-modules need not be.

Refs

  • Hörmander, An Introduction to Complex Analysis in Several Variables
  • Gunning–Rossi, Analytic Functions of Several Complex Variables
  • Krantz, Function Theory of Several Complex Variables
  • Grauert–Remmert, Coherent Analytic Sheaves

Next

Foundations · \(\bar\partial\)-Neumann & Kohn Estimates · Stein & Oka Principle · CR Geometry & \(\bar\partial_b\) · Analytic Methods & \(L^2\) · Complex geometry · Complex analysis · Commutative algebra · AG foundations