Topic course in algebraic geometry:
GIT and moduli of vector bundles

Description

Geometric invariant theory (GIT) is a general method to construct quotients for an algebraic variety (or scheme) acted on by a group. One of the original motivations for developing GIT is to provide constructions of moduli spaces of various types of objects in algebraic geometry. In the first part of this course, we will begin with a study of GIT from a more general view point and then look at several examples of moduli problems in algebraic geometry where GIT can be applied. The second part of the course is then devoted to study in details the construction and properties of the moduli space M(r,d) of vector bundles on curves and its variants (if time permits).

Part I: Geometric invariant theory (GIT)

  • Algebraic groups and their actions
  • Affine GIT
  • Projective GIT
  • Hilbert-Mumford criterion
  • Moduli problems and examples

Part II: Moduli spaces of vector bundles on curves

  • Semi-stability of vector bundles on curves
  • The Quot scheme
  • Construction of M(r,d)
  • Properties of M(r,d)
  • Cohomology of M(r,d)
  • Variants of M(r,d)

References

We will mainly follow the lecture notes by V. Hoskins and "Lectures on Vector Bundles by J. Le Potier. You may also find the following list of references useful:
  • "Lectures on Invariant Theory" by I. Dolgachev.
  • "Introduction to Moduli Problems and Orbit Spaces" by P. E. Newstead.
  • "Introduction to the Theory of Moduli" by D. Mumford and K. Suominen.
  • "An Introduction to Invariants and Moduli" by S. Mukai.
  • "Geometric Invariant Theory" by D. Mumford, J. Fogarty, F. Kirwan.
  • "The Geometry of Moduli Spaces of Sheaves" by D. Huybrechts and M. Lehn.

Course information

Prerequisites: I will assume some basics of algebraic geometry (e.g. chapter 2 and 3 of Hartshorne "Algebraic Geometry")
Final grade: The final grade will be given based on homework and a take-home exam.
Medium of instruction: English
Time: Wednesday (even-numbered weeks) 8:00-9:50, Friday (all weeks) 13:00-14:50
Location: Teaching Building No.2 (二教) Room 421

Course summary

  • Lecture 1: Introduction to moduli problems; Relation between moduli and quotients
  • Lecture 2: Moduli of endomorphisms; Introduction to algebraic groups and group actions
  • Lecture 3: Tori; Orbits and stabilizers
  • Lecture 4: Quotients: categorical, good, geometric quotients, principal bundles
  • Lecture 5: Reductive groups: reductivity, linear reductivity, geometric reductivity
  • Lecture 6: Finite generation of invariant subalgebras
  • Lecture 7: Invariant theory and affine quotient
  • Lecture 8: Linearization and GIT: proof of the existence of GIT quotient
  • Lecture 9: More on GIT and examples
  • Lecture 10: Hilbert-Mumford criterion I: proof
  • Lecture 11: Hilbert-Mumford criterion II: some examples such as the Grassmannians
  • Lecture 12: Examples of moduli problems: projective hypersurfaces
  • Lecture 13: Examples of moduli problems: representations of quivers
  • Lecture 14: Coherent sheaves on curves and slope stability
  • Lecture 15: Properties of (semi)stable bundles; Jordan-Holder and Harder-Narasimhan filtration
  • Lecture 16: Boundedness of semistable bundles and Castelnuovo-Mumford criterion
  • Lecture 17: Construction of Quot schemes
  • Lecture 18: Construction of M(r,d) I: analysis of GIT stability
  • Lecture 19: Construction of M(r,d) II: comparison of GIT stability and slope stability
  • Lecture 20: "Stable pairs, linear systems and the Verlinde formula" by M. Thaddeus
  • Lecture 21: Existence of universal bundles, irreducibility and rationality of M(r,d)
  • Lecture 22: Introduction to deformation theory: tangent-obstruction theory
  • Lecture 23: Examples of tangent-obstruction theories. Smoothness of M(r,d)