Complex Analytic Spaces

1. Sheaves, manifolds, complex manifolds, holomorphic functions, Cauchy formula in one and many variables. Hartogs’ theorem.
2. Limits and colimits. Germs of continuous, smooth and holomorphic functions.
3. Weierstrass preparation theorem. Weierstrass divisibility theorem. Noetherian rings. Lasker-Noether theorem (Noetherianity of germs of holomorphic functions).
4. Complex analytic sets and complex analytic varieties. Germs of complex subvarieties. Local parametrization of germs of complex analytic varieties (Noether normalization lemma).
5. Remmert’s proper mapping theorem. Remmert-Stein extension theorem. Chow theorem.
6. Coherent sheaves in analytic category. Oka coherence theorem.
7. Normal complex analytic varieties. Normalization.
8. Normal families of holomorphic functions. Montel theorem. Montel sheaves and Montel spaces. Finiteness of cohomology of Montel sheaves on compact according to Grothendieck.

The definite textbook on the subject is Demailly’s “Complex analytic and differential geometry”,
A. Grothendieck, Theoremes de finitude pour la cohomologie des faisceaux, Bull. Soc. Math. France 84 (1956), 1-7,
Gunning, R.C., Rossi, H. [1965] Analytic functions of several complex variables.
Grauert, H., Remmert, R. [1984] Coherent analytic sheaves.