1. Manifolds and vector bundles
1.1 Manifolds-projective algebraic manifolds, Grassnnian manifodls.
1.2 Vector bundles-tngent bundle bundle, universal bundle
1.3 Almost complex manifolds and the d-bar operator-almost complex structure, integrable structure, Newlander-Nirenberg’ Theorem.
2. Sheaf theory
2.1 Presheaves and sheaves-locally free sheaf
2.2 Resolutions of sheaves
2.3 Cohomology theory-de Rham theorem, Cech cohomology.
3. Differential Geometry
3.1 Hermitian differential geometry- connection, curvature tensor
3.2 The canonical connection and curvature of a Hermitian holomorphic vector bundle.
3.3 Chern classes of differentiable vector bundles-introduction of Chern forms
3.4 Complex line bundles
4. Elliptic Operator Theory
4.1 Sobolev Spaces-Sobolev norm, Sobolev embedding theorem
4.2 Differential Operators-k-symbol
4.3 Pseudodifferential operators
4.4 A parametrix for elliptic differential operators
4.5 Elliptic complexes-Laplace operator, Green operator, harmonic forms, Euler characteristic,
Riemann-Rock-Hirzebruch Theorem
5. Compact complex manifolds
5.1 Hermitian exterior algebra on a Hermitian vector space
5.2 Harmonic theory on compact manifolds-star operator, complex Laplace operator, Poincare duality, Serre duality
5.3 Representation of sl(2,C) on Hermitian exterior algebras-Lefscheta decomposition theorem, primitive decomposition
5.4 Differential operators on a Kahler manifold-strong
5.5 The Hodge decomposition theorem on compact Kahler manifolds
5.6 The Hodge-Riemann bilinear relations on a Kahler manifold-Hodge filtration
6. Kodaira’s projective embedding theorem
6.1 Hodge manifolds
6.2 Kodaira vanishing theorem
6.3 Quadratic transformations
6.4 Kodaira’s embedding theorem.