MATH 216 : 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.

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Announcement: The course will begin on Monday, January 14, 2013.

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

• Lecture: MWF 11-11:50, MS 6201

• Quiz section: None

• Office Hours: By appointment.

• Textbook: Mukai's "An introduction to invariants and moduli" (Cambridge, $70.00) 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, $164.73) by Mumford-Fogarty-Kirwan can be intimidating as an introduction, but it contains a lot of fascinating mathematics.

• Prerequisite: The main prerequisite is a first course on algebraic geometry, so that you have seen 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. If you are learning algebraic geometry this quarter in Math 214A, then you may get something out of this course, but you'll have to put in some effort.

• Grading: This is a topics course, so I am planning a somewhat informal grading scheme. Basically, the base grade will be B if you actually show up to a significant number of classes, adjusted upwards according to whether you turn in any homework. There will be one or two homework sheets, and no final or midterm.

• Homework 1 (due February 20, 2012).