Complex algebraic geometry

Algebraic geometry can be studied two different ways.
You can deduce everything from commutative algebra,
as it was done in classical Italian school of algebraic
geometry. This is an elementary approach, but not
intuitive and using a lot of exhausting work.
Instead, as Hodge suggested, the foundation
of algebraic geometry can be based on topology
and differential geometry. The main tool of this
approach is the theory of harmonic forms on Riemannian
manifolds, known as "Hodge theory"; in addition,
one uses complex analysis, differential geometry
and homological algebra. This approach (also called
"Hodge theory") is more straightforward and intuitive,
but it takes much preliminary work around
topology and analysis. Also, Hodge theory
works only in characteristic 0.

This course is based on differential geometry
(manifolds, bundles, connections, de Rham algebra,
Stokes' theorem) algebraic topology (de Rham cohomology,
intersection theory, Poincare duality) and complex
analysis (Taylor decomposition of holomorphic functions).
I would assume a few results of spectral theory
of Laplacian operators without proof.

Program
(the list of possible topics for the course)

1. Complex structures, almost complex structures,
Hodge decomposition on differential forms.

2. Almost complex manifolds and their integrability,
Newlander-Nirenberg theorem for real analytic manifolds.

4. Hermitian metrics, Kahler manifolds, examples
and main properties of Kahler manifolds. Fubini-Study metrics.
Kahler metrics on homogeneous complex manifolds.

5. Levi-Civita connection on Kahler manifolds and its properties.

6. Supersymmetry on Kahler manifolds and its applications:
Kahler identities, Hodge decomposition, Lefschetz theorem,
sl(2)-triples and Lefschetz decomposition on cohomology.

7. Currents and generalized functions. Integral kernels.
Cauchy kernel.

8. Poincare-Dolbeault-Grothendieck lemma.
Dolbeault cohomology. Geometric interpretation
of the Hodge decomposition and its applications.

9. Holomorphic differential forms and their properties.
Birational maps. Blow-ups. Invariance of holomorphic
differential forms under birational maps. Canonical bundle
and its properties.

10. Holomorphic bundles. Chern connection, its existence
and uniqueness, its curvature. Line bundles, exponential
exact sequence, first Chern class and its properties.

11. Supersymmetry algebra of a Kahler manifold, its
action on differential forms with coefficients in a bundle.
Kodaira-Nakano identities. Kodaira-Nakano vanishing theorem.

12. Globally generated, ample and very ample bundles.
Projective embeddings. Kodaira embedding theorem.
Algebraic dimension of complex manifolds. Moishezon,
complex non-algebraic and non-Kahler manifolds.

13. (*) Abelian manifolds and complex tori.
Albanese map and its properties. Holomorphic
differentials on Riemann surfaces.

14. (*) Calabi-Yau theorem, Calabi-Yau manifolds,
Monge-Ampere equation, classification of Riemannian
holonomies and its applications.

Last two subjects will be considered
if time permits only.

Literature
Lectures on Kahler geometry, Andrei Moroianu
http://moroianu.perso.math.cnrs.fr/tex/kg.pdf

Complex analytic and differential geometry, J.-P. Demailly
http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

Lectures on Kahler manifolds, W. Ballmann
http://people.mpim-bonn.mpg.de/hwbllmnn/notes.html

C. Voisin, ``Hodge theory''.

D. Huybrechts, ``Complex Geometry - An Introduction''

A. Besse, ``Einstein manifolds''.