Complex and symplectic manifolds arise in several different situations, from the study of complex polynomials to mechanics and string theory. Both concepts are central to two major branches of the mathematics research area "geometry". At first sight, these two concepts share little in common, but there is a rich interplay between them which leads to powerful results, specially on Kahler manifolds (manifolds which admit a pair of compatible complex and symplectic structure).

This course is a follow up of the courses on Riemann surfaces and on Differentiable manifolds. We will study compact complex manifolds, complex and holomorphic vector bundles and will pay special attention to Kähler manifolds.

The point of view followed will be differential geometric.

The main concepts we will study are:

  • complex and Kähler manifolds
  • complex and holomorphic vector bundles
  • line bundles and divisors
  • Dolbeault cohomology

The main aims of this course are to state and prove the following results:

  • Hodge Theorem for Kahler manifolds
  • Kodaira vanishing theorem
  • Kodaira embedding theorem


  • Differentiable manifolds (tangent and cotangent bundles, vector bundles, flows of vector fields, Lie derivative, differential forms, exterior derivative and de Rham cohomology, as in chapters 1 and 4 of Warner's book "Foundations of Differentiable manifolds")
  • Riemann surfaces (sheaves, sheaf cohomology and basic properties of sheaf cohomology)