Foth Philip. Complex manifolds, vector bundles and Hodge theory

Boston: Birkhauser, 1998. — 268 p.Vector bundles arise in geometry in several contexts. One may remember from the study of smooth manifolds that the notion of tangent bundle inevitably appeared as a powerful tool of differential geometry.
If the dimension of a manifold M is k then the dimension of the total space of the tangent bundle to M is twice as big. The first and simple example arises when we take M = R k . Here the tangent bundle is just the direct product of two copies of R k . So, T R k = {(x, y); x, y ∈ R k }.Holomorphic vector bundles
Vector bundles over smooth manifolds
Complex manifolds
Holomorphic line bundles
Divisors on Riemann surfaces
Line bundles over complex manifolds
Intersection of curves inside a surface
Theta function and Picard group
Cohomology of vector bundles
Cech cohomology for vector bundles
Extensions of vector bundles
Cohomology of projective space
Chern classes of complex vector bundles
Construction of the Chern character
Riemann-Roch-Hirzebruch theorem
Connections, curvature and Chern-Weil
The case of holomorphic vector bundles
Riemann-Roch-Hirzebruch theorem for CP n
RRH for a hypersurface in projective space
Applications of Riemann-Roch-Hirzebruch
Dolbeault cohomology
Grothendieck group
Algebraic bundles over CP n