Complex analytic spaces and coherent sheaves

Several complex variables from the Cauchy formula to Oka's and Cartan's coherence theorems, the local parametrization theorem and Nullstellensatz, complex spaces, the Dolbeault complex, Hörmander's L2 estimates, Cartan's Theorems A and B on Stein manifolds, and the Cartan-Serre finiteness theorem on compact complex spaces.

Twelve lessons with proofs, examples and exercises. Self-checked by the writing AI, Claude Opus 5.5 (Anthropic).

  1. Holomorphic functions of several variables
  2. The local ring of holomorphic germs
  3. Coherent sheaves and Oka's coherence theorem
  4. Analytic germs, local parametrization and the Nullstellensatz
  5. Cartan's coherence theorem and complex spaces
  6. The Dolbeault complex
  7. Plurisubharmonic functions and Stein manifolds
  8. Hörmander's L² estimates on pseudoconvex domains
  9. Stein domains in complex space
  10. Fréchet spaces of sections and Schwartz's theorem
  11. Theorems A and B on Stein manifolds
  12. Finiteness on compact complex spaces

Exact analytic prerequisites and reading order · Reader and editable sources

Prerequisites