Introduction to Geometric Invariant Theory

20-24 January 2020

Title of the course: Introduction to Geometric Invariant Theory
Instructor: Dr. Yoshinori Hashimoto
Institution: Tokyo Institute of Technology
Dates: 20-24 January 2020
Prerequisites: Basic complex algebraic geometry
Level: Graduate and Advanced undergraduate
Abstract: Suppose that a reductive algebraic group G acts on a projective variety X. Geometric Invariant Theory (GIT), initiated by Mumford, is a method of taking a quotient X/G in the category of varieties. It has many important applications in the theory of moduli and is also related to problems in differential geometry. This minicourse aims to be a rapid introduction to this topic; it will be mostly about the construction of the GIT quotient and the Hilbert—Mumford criterion, but related differential-geometric topics will also be mentioned.
Textbook and References:
1. Dolgachev, I. Lectures on invariant theory. London Mathematical
Society Lecture Note Series, 296. Cambridge University Press, Cambridge, 2003. xvi+220 pp. ISBN: 0-521-52548-9

  1. Mukai, S. An introduction to invariants and moduli. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003. xx+503 pp. ISBN: 0-521-80906-1
  2. Huybrechts, D.; Lehn, M. The geometry of moduli spaces of sheaves. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010. xviii+325 pp. ISBN: 978-0-521-13420-0
  3. Chriss, N.; Ginzburg, V. Representation theory and complex geometry. Reprint of the 1997 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2010. x+495 pp. ISBN: 978-0-8176-4937-1
  4. Thomas, R. P. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in differential geometry. Vol. X, 221–273, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006. https://arxiv.org/abs/math/0512411
  5. Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. x+440 pp. ISBN: 0-19-853553-8
  6. Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN: 3-540-56963-4
    Language: EN