Math 739 — Topics in Algebra

Spring 2003

Instructor: Dan Zaffran

Time: TR 10:10-11:25

Room: MT 206

Topic: Complex Geometry

Complex numbers, although painfully discovered as a tool to solve cubic equations, have proved to be fundamental mathematical objects. One distant consequence of their nice properties lies in the beauty of complex geometry, which is differential geometry in the category of holomorphic manifolds and holomorphic maps. Furthermore, that theory interacts nicely with other areas. Roger Penrose pointed that out in a slightly provocative way, by entitling an article "The complex geometry of the natural world".

The course theme is the study of algebraic and non-algebraic complex manifolds. Beyond general facts, possible topics include (an introduction to):

• automorphism groups,
• classification results for compact manifolds,
• complex spaces (i.e., "manifolds with singularities"),
• holomorphic foliations,
• deformations.

Prerequisites:

• (Rather) non-negotiable: smooth differential geometry of manifolds. Some acquaintance with fiber bundles, Lie groups, one-variable complex analysis, algebraic topology.
• (Highly) negotiable: a good background would be Math 722 (several complex variables, sheaves and cohomology), but I may give at least crash courses on these topics, and probably much more...