To fix the date of the exam for the +3 CFU contact me via email.
Exercises for Algant students: do two exercises from the file.
Lecture 1, November 15 (still 6CFU).
Motivation to Cohomology: the Mittag-Leffler problem [S, page 36].
Čech cohomology: definition and examples.
Lecture 2, November 16 (still 6CFU).
Comparison theorems between Čech cohomology and other cohomologies.
Fine sheaves and Doulbeault theorem on (p,q)-forms.
Lecture 3, November 18 (still 6CFU).
Picard group and H^1. Exponential sequence.
Examples.
Lecture 4, November 22.
Cohomology and Picard group of projective space (still 6cfu).
Cohomology of hypersurfaces (still 6cfu).
Introduction to the second part of the course +3CFU.
Lecture 6, November 23 (+3CFU).
Riemannian manifolds and volume forms [S, Sec. 11].
Linear algebra of Hermitian forms [S, Sec. 12] or [V, 3.1].
Lecture 7, November 25 (+3CFU).
Hermitian forms, local computations.
Volume form of a Hermitian manifold [V, 3.1.3].
The Fubini-Study metric on the projective space [H, page 117].
Lecture 8, November 29 (+3CFU).
Kähler manifolds: basic properties, first examples and non-examples.
Kähler metrics are locally flat up to the first order.
Lecture 9, November 30 (+3CFU).
Levi-Civita connection.
Dolbeault complex of a holomorphic vector bundle.
Chern connections.
Lecture 10, December 2 (+3CFU).
Connections on a Kähler manifold.
Curvature form of a line bundle and first Chern class.
Lecture 11, December 9 (+3CFU).
Weil and Cartier divisors and relation to line bundles.
Lecture 12, December 13 (+3CFU).
Fundamental class of a divisor and applications.
Lecture 13, December 14 (+3CFU).
Real harmonic theory
Lecture 14, December 16 (+3CFU).
Complex harmonic theory.
Hodge decomposition and consequences.
Lecture 15, December 20 (+3CFU).
Maps to the projective space.
Blow-ups.
Lecture 16, December 21 (+3CFU).
-Properties of blow-ups.
-Kodaira's embedding theorem.
References:
• [H] D. Huybrechts, Complex Geometry, an introduction. Berlin Springer-Verlag 2005.
• [S] C. Schnell, Lecture notes online "Complex Manifolds".
• [V] C. Voisin, Hodge Theory and Complex Algebraic Geometry, I. Cambridge University Press 2002.