Complex analysis in C^n for n ≥ 2, where the phenomena diverge fundamentally from the single-variable case. Hartogs 1906 — for n ≥ 2, a holomorphic function on the 'Hartogs figure' (a polydisk with a smaller polydisk removed) extends…
several-complex-variables
Polydisk D^n and its distinguished boundary T^n
The polydisk D^n = {z ∈ C^n : |zᵢ| < 1 for all i} is the product of n copies of the unit disk. Its topological boundary ∂D^n = {z : max…
Hartogs extension (codim-2 singularities auto-extend for n ≥ 2)
Hartogs 1906 (Math. Ann. 62). For n ≥ 2, a holomorphic function on the Hartogs figure H_{n,ε} = {z : |z₁| < 1, |z_j| < ε for j ≥ 2} ∪ {z :…
Domain of holomorphy: holomorphically convex ⇔ pseudoconvex
A domain Ω ⊂ C^n is a 'domain of holomorphy' if there is a holomorphic f on Ω with no holomorphic extension past any boundary point. …
Wirtinger ∂̄-residual ≡ 0 on f(z₁,z₂) = z₁²z₂³
Exact sympy verification of the Cauchy-Riemann system in two complex variables for the monomial f(z₁, z₂) = z₁² z₂³ on C² with Wirtinger…
Bergman kernel K_D(z,w̄) = 1/(π(1-zw̄)²); K_D(0,0̄) = 1/π
The Bergman reproducing kernel of the open unit disk D = {|z| < 1} on the Hilbert space A²(D) = L²(D) ∩ Hol(D) is K_D(z, w̄) = 1/(π(1 −…
Σ z₁ʲz₂ᵏ/4^{j+k} = 16/((z₁-4)(z₂-4)); pins at (1,1), (0,2)
Double geometric series in two complex variables — the canonical Hartogs-series example that converges on the full bidisk D²(4) = {|z₁|,…
Hartogs extension theorem
Hartogs 1906: in ℂⁿ for n ≥ 2, holomorphic function on shell extends to interior. Singularities cannot be isolated in complex dim ≥ 2…
Domain of holomorphy
Open Ω ⊂ ℂⁿ such that some holomorphic function on Ω cannot be extended past any boundary point. Cartan-Thullen: Ω is domain-of-holomorphy…
Plurisubharmonic & pseudoconvex
u plurisubharmonic: subharmonic on every complex line. Levi pseudoconvex domain: boundary admits exhausting plurisubharmonic function. …
Dolbeault cohomology
∂̄-operator on (p,q)-forms; Dolbeault cohomology H^{p,q}(M) = ker(∂̄)/im(∂̄). Hodge decomposition for compact Kähler. Cech-Dolbeault…
Oka's coherence theorem
Oka 1950: sheaf of holomorphic functions on complex manifold is coherent. Cartan A,B theorems: H^q(X,F) = 0 for q ≥ 1 on Stein manifold. …
Bergman kernel
K(z,w) reproducing kernel for L²-holomorphic functions on Ω. K(z,z) = sup{|f(z)|² : ||f||₂ = 1}. Bergman metric ds² = ∂²log K/∂z_i∂z̄_j…
Hartogs extension (1906)
Hartogs 1906: holomorphic f on Omega minus K extends to all of Omega for K compact, n>=2; no isolated singularities in higher-dim…
Plurisubharmonic / pseudoconvex (SCV)
Oka 1953 + Bremermann 1954: domain Omega is domain-of-holomorphy iff pseudoconvex (Levi-form positive-semidefinite on boundary);…
d-bar equation (Hormander 1965)
Hormander 1965: dbar u = f on pseudoconvex domain has solution with L^2 estimate ||u||_phi <= ||f||_phi for plurisubharmonic weight phi;…
Oka-Cartan coherent sheaves
Oka 1950 + Cartan 1944-1953: coherent analytic sheaves on Stein manifolds satisfy H^q(X, F)=0 for q>=1 (Theorem B); foundation of…
Kodaira vanishing + embedding
Kodaira 1953-1954 (Fields 1954): vanishing H^q(X, K_X otimes L)=0 for L positive q>=1; embedding theorem: X compact-Kahler + Hodge ->…
Bergman kernel / Szego projection
Bergman 1922 reproducing kernel K(z,w) of L^2-holomorphic on Omega; Szego 1921 projection on bdry; Fefferman 1974 asymptotic at…
Hartogs (1906)
F Hartogs 1906 simultaneous-continuation; modern modern foundational text + analytic continuation + Hartogs phenomenon n>1 separator.
Oka (1939)
K Oka 1939 + 1953 coherent-sheaves; modern modern foundational text + Cartan-Thullen + Oka principle + Stein theorem 1956.
Levi (1911)
E Levi 1911 pseudo-convex hypersurface; modern modern foundational text + ∂̄-problem Hörmander 1965 + Bergman kernel.
Dolbeault (1953)
P Dolbeault 1953 ∂̄-cohomology; modern modern foundational text + Dolbeault-Grothendieck + Kodaira-vanishing-thm.
Bergman kernel (1933)
S Bergman 1933 reproducing-kernel; modern modern foundational text + Bergman-metric + Bergman-projection + complex-geometry.
Fefferman (1976)
C Fefferman 1976 (Fields 1978) biholomorphic-extension; modern modern foundational text + complex Monge-Ampère + Cheng-Yau.