28th Autumn School in Algebraic Geometry

Equivariant intersection theory

Lukecin, Poland, September 11th - September 17th, 2005

Teachers: Andrew Kresch (Warwick, UK) and Angelo Vistoli (Bologna, Italy)

Program: The subject of our lectures will be equivariant intersection theory. To any smooth complex algebraic variety X one associates two important rings, the Chow ring A(X) and the integral cohomology ring H(X). If G is a linear algebraic group acting on a complex smooth variety X, there are variants of these, called the equivariant Chow ring A_G(X) and the equivariant cohomology ring H_G(X). We will give an elementary introduction to the theory, focusing on examples. We will describe applications in algebraic geometry, to the calculations of Chow rings of important moduli spaces, to questions in representation theory, and to enumerative geometry and Gromov-Witten theory.

Preliminary reading list:

  • W. Fulton, Intersection theory, Springer-Verlag, Berlin (1984).
  • A. Borel et al., Seminar on Transformation Groups, Ann. of Math. Studies 46, Princeton University Press, Princeton, 1960.
  • Burt Totaro, The Chow ring of a classifying space, Algebraic K-theory (Seattle, WA, 1997), Amer. Math. Soc., Providence, RI, 1999, pp. 249-281
  • Dan Edidin and William Graham, Equivariant intersection theory , Invent. Math. 131 (1998), 595-634
  • M. Brion, Equivariant cohomology and equivariant intersection theory , in "Representation theory and algebraic geometry", Kluwer (1998), pp. 1-37
  • K. Behrend, Localization and Gromov-Witten Invariants, in "Quantum cohomology" (Cetraro 1997) Lecture Notes in Mathematics 1776 (2002), pp. 3-38
  • R. Pandharipande, Equivariant Chow rings of O(k), SO(2k+1), and SO(4), J. Reine Angew. Math. 496, (1998), 131-148.
  • R. Pandharipande, The Chow ring of the non-linear Grassmannian, J. Alg. Geom. 7 (1998), 123-140
  • M. Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994), 335-368, Progr. Math. 129, Birkhauser Boston, MA, 1995
  • G. Ellingsrud, S.A. Stromme, Bott's formula and enumerative geometry, J. Amer. Math. Soc. 9 (1996), no. 1, 175-193
  • J. Carrell, K. Kaveh and V. Puppe, Vector Fields, Torus Actions and Equivariant Cohomology, preprint (15 pages), 2005

    Organizers: Jaroslaw Buczynski and Adrian Langer.

    The school took place in a Warsaw University pension in Lukecin (look here for more information), on Western part of Polish Baltic sea shore (see a map). The school was financially supported by Institute of Mathematics of Warsaw University and by Polish State Committee for Scientific Research.

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