Topics in Complex Geometry

While noncommutative geometry is a well developed subject, we have so far only glimpses of a noncommutative complex geometry. This year's topics in complex geometry will be centered around ``complex and holomorphic noncommutative geometry". There is no general theory as yet, and the subject is driven mainly by examples: holomorphic structures on noncommutative tori and their relations to elliptic curves, holomorphic geometry of noncommutative projective line from quantum groups.

Outlines and Lecture notes:

After a quick review of basic tools of classical complex manifold theory, the main part of the course will be devoted to a detailed study of the noncommutative projective line. The following topics will be studied:

  • Almost complex structures and complex structures,
  • Hodge theory of complex manifolds,
  • quantum SU_2, and quantum sphere,
  • Differential calculus on the quantum sphere and holomorphic structures,
  • Holomorphic line bundles on quantum spheres