## MATH 216A : Geometric Invariant Theory

• Course description: The construction of quotients of algebraic varieties by group actions, with applications to moduli spaces and the topology of quotient varieties.
Examples of group actions in algebraic geometry. Complex reductive groups and Hilbert's theorem on finite generation of rings of invariants. Quotients of projective varieties; stable and semistable points. The Hilbert-Mumford criterion. The cohomology of quotient spaces in some examples. The moduli space of vector bundles on a curve. The Narasimhan-Seshadri theorem, relating stability for vector bundles on a curve with the existence of flat Hermitian metrics. The moduli space of curves and its compactification by stable curves.

• Instructor: Burt Totaro, totaro@math.ucla.edu, MS 6136

• Lecture: MWF 1:00-1:50, MS 6201

• Office Hours: By appointment.

• Textbook: Mukai's "An introduction to invariants and moduli" (Cambridge, \$82.00) is the main book for the class. This is a very readable book that covers the theory and a key example (the moduli space of vector bundles on a curve) in detail. It is definitely worth buying. The original book "Geometric invariant theory" (Springer, \$209.00) by Mumford-Fogarty-Kirwan can be intimidating as an introduction, but it contains a lot of fascinating mathematics.

• Prerequisite: The prerequisite is a solid introduction to algebraic geometry, which in UCLA terms means Math 214A and 214B. So you must be familiar with things like affine and projective varieties, the genus of a curve, and so on. However, I will make simplifying assumptions, and work with explicit examples, to try to be more understandable. For example, I will only consider varieties over the complex numbers.