Complex Analysis: Multiple Variables
Theory of complex-analytic spaces is parallel to the complex algebraic geometry and commutative algebra.
Almost all local notions of algebraic geometry have their analogues in complex analysis, but the proofs are really different. The highest pinnacle of this theory is Chow theorem which claims that complex subvarieties of a complex projective space are algebraic.
This is a course for students who mastered complex analysis in one variable, basic topology and theory of smooth manifolds, and want to know about complex analysis of multiple variables.
The main subject of these lectures is the local parametrization of complex varieties (analogue of Noether lemma).
0. Sheaves, manifolds, complex manifolds, holomorphic functions, Cauchy formula in one and many variables. Hartogs’ theorem.
1. Limits and colimits. Germs of continuous, smooth and holomorphic functions.
2. Weierstrass preparation theorem. Weierstrass divisibility theorem. Noetherian rings. Lasker-Noether theorem (Noetherianity of germs of holomorphic functions).
3. Complex analytic sets and complex analytic varieties. Germs of complex subvarieties. Local parametrization of germs of complex analytic varieties (Noether normalization lemma).
4. Remmert’s proper mapping theorem. Remmert-Stein extension theorem. Chow theorem.
5* Coherent sheaves in analytic category. Oka coherence theorem.
6* Normal complex analytic varieties. Normalization.
7* Normal families of holomorphic functions. Montel theorem. Montel sheaves and Montel spaces. Finiteness of cohomology of Montel sheaves on compact according to Grothendieck.
The last 3 topics (marked with *) are much less elementary; they will be included only if time permits and the students are able to master the rest.
Demailly’s., – Complex analytic and differential geometry”,https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
A. Grothendieck., – Theoremes de finitude pour la cohomologie des faisceaux, Bull. Soc. Math. France 84 (1956), 1-7, http://archive.numdam.org/article/BSMF_1956__84__1_0.pdf
Gunning, R.C., Rossi, H., – Analytic functions of several complex variables, 1965.
Grauert, H., Remmert, R., – Coherent analytic sheaves, 1984.