28th Autumn School in Algebraic Geometry
Equivariant intersection theory
Lukecin, Poland, September 11th - September 17th, 2005
Andrew Kresch (Warwick, UK) and Angelo Vistoli (Bologna, Italy)
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
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.
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
Equivariant cohomology and equivariant intersection theory , in
"Representation theory and algebraic geometry", Kluwer (1998), pp. 1-37
Localization and Gromov-Witten Invariants, in "Quantum
cohomology" (Cetraro 1997) Lecture Notes in Mathematics 1776 (2002), pp.
R. Pandharipande, Equivariant Chow rings of O(k), SO(2k+1), and SO(4), J.
Reine Angew. Math. 496, (1998), 131-148.
The Chow ring of the non-linear Grassmannian, J. Alg.
Geom. 7 (1998), 123-140
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
The school was financially supported by Institute of
Mathematics of Warsaw University and by Polish State Committee
for Scientific Research.