To fix the date of the exam contact me via email.
Exercises for Algant students: do one or two exercises of the file.
Lecture 11, December 19.
Hodge theory: real and harmonic forms, decomposition theorem and some applications [S. Sec. 18, 19, 23, 35].
Lecture 10, December 18.
Kodaira embedding theorem [S. Sec. 32-34].
Lecture 9, December 13.
Chern class of line bundles [S. Sec. 29-30].
Maps to projective space [S, Sect 32].
Lecture 8, December 12.
Kähler metrics and connections [V, 3.2.2].
Lecture 7, December 11.
Dolbeault complex of a holomorphic vector bundle [V, 2.3.3].
The Chern connection [V 4.2.1].
Lecture 6, December 5.
Volume form of a Hermitian manifold [V, 3.1.3].
Fubini-Study metric on the projective space [H, page 117].
The Levi-Civita connection ([V, 3.2.1]).
Lecture 5, December 4.
Linear algebra of Hermitian forms.
Hermitian forms, local computations and Kähler forms.
See [S, Sec. 12] or [V, 3.1]
Lecture 4, November 28.
Cohomology of projective hypersurfaces II.
Riemannian manifolds and volume forms [S, Sec. 11].
Lecture 3, November 27.
Picard group and H^1. Exponential sequence.
Cohomology of projective hypersurfaces I.
Lecture 2, November 21.
Comparison theorems between Čech cohomology and other cohomologies.
Fine sheaves and Doulbeault theorem on (p,q)-forms.
Lecture 1, November 20.
Motivation to Cohomology: the Mittag-Leffler problem [S, page 36].
Čech cohomology: definition and examples.
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.