When a reductive linear algebraic group G acts on an algebraic variety X, its topological quotient X/G is not necessarily an algebraic variety. Geometric Invariant Theory (GIT) initiated by Mumford, on the other hand, gives an appropriate definition of X/G as an algebraic variety. This course covers introductory materials in GIT. While GIT plays a fundamental role in constructing moduli spaces, we shall only touch on elementary examples of moduli spaces in this course. The lecture plans are only approximate.
The aim of this course is to understand various fundamental definitions and constructions in GIT, and to understand the important notion of stability. This course can be considered as a prequel to Advanced topics in Geometry F whose final aim is to prove the Kempf-Ness theorem, which states that a point in X is polystable in the sense of GIT if and only if it is a zero of the moment map. That said, this course should serve as a stand-alone introduction to GIT.
Students will learn: reductive linear algebraic groups and their action to algebraic varieties; GIT quotient and the concept of stability; Hilbert-Mumford criterion, a criterion of stability, and use it to determine stability in some elementary examples.
Geometric Invariant Theory, reductive linear algebraic group, stability, Hilbert-Mumford criterion
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||✔ Practical and/or problem-solving skills|
Standard lectures, problems for the homework assignments will be given during the course.
|Course schedule||Required learning|
|Class 1||Introduction and review of basic complex algebraic geometry||Details will be provided during each class session.|
|Class 2||Linear algebraic groups and reductive linear algebraic groups||Details will be provided during each class session.|
|Class 3||Affine GIT quotients||Details will be provided during each class session.|
|Class 4||Categorical quotients||Details will be provided during each class session.|
|Class 5||Linearisation of actions of linear algebraic groups||Details will be provided during each class session.|
|Class 6||Stability||Details will be provided during each class session.|
|Class 7||Hilbert-Mumford criterion||Details will be provided during each class session.|
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
None, but typeset notes will be distributed as appropriate.
I. Dolgachev, Lectures on invariant theory. London Mathematical Society Lecture Note Series, 296. Cambridge University Press, Cambridge, 2003. xvi+220 pp. ISBN: 0-521-52548-9
D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN: 3-540-56963-4
P. Newstead. Introduction to moduli problems and orbit spaces. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 51. Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978. vi+183 pp. ISBN: 0-387-08851-2
R. Thomas. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in differential geometry. Vol. X, 221--273, Int. Press, Somerville, MA, 2006. (available at https://arxiv.org/abs/math/0512411)
Graded by homework assignments.
Students are assumed to be familiar with the basics of complex algebraic geometry, such as affine varieties, projective varieties, Zariski topology, ample line bundles, and related topics in commutative algebra. On the other hand, supplementary explanations will be given during the course for these basic concepts as much as possible; in particular, we shall very quickly recall the definitions of affine varieties and projective varieties (or Spec and Proj) in the first lecture.
Not specifically, but extra discussions can be arranged by email or after each class.