Introduction to Complex Geometry

Prerequisite: Complex Analysis

Hartogs’ theorem in the polydisk. Hartogs’ extension theorem. Holomorphically convex domains. Pseudoconvexity. Relationships between holomorphic convexity and pseudoconvexity. Dolbeault’s Lemma  and cohomology. Cousin’s problem. Sheaves, Cech cohomology with coefficients in sheaves. Exact short and long sequences. Acyclic resolutions. Stein varieties. Theorems A and B of Cartan (statements) and applications. Kaehler varieties. Hodge decomposition. Relationship between line bundles and divisors. First Chern class.  Serre Duality.

References:
CHABAT, S. – Introduction `a l’Analyse Complexe, Vol.2. MIR, 1990.
GUNNING, R. – Introduction to Holomorphic Functions of Several Variables, Vol. I, II e III Belmont, Ed. Wadsworth, 1990.
FRITZSCHE, K., GRAUERT, H. – From Holomorphic Functions to Complex Manifolds. Graduate Texts in Mathematics, 213. Springer-Verlag, New York, 2002.
VOISIN, C.- Hodge Theory and Complex Algebraic Geometry, I.