TOWARDS AN INTERSECTION CHOW COHOMOLOGY THEORY FOR GIT QUOTIENTS
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Springer Science+Business Media New York (2020)
TOWARDS AN INTERSECTION CHOW COHOMOLOGY THEORY FOR GIT QUOTIENTS DAN EDIDIN∗
MATTHEW SATRIANO∗∗
Department of Mathematics University of Missouri Columbia MO 65211, USA
University of Waterloo Department of Pure Mathematics Waterloo, Ontario Canada N2L 3G1
[email protected]
[email protected]
Abstract. We study the Fulton–MacPherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are ´etale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with Q-coefficients, every operational class can be represented by a topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Our methods also allow us to prove that if X is the good moduli space of a properly stable, smooth, Artin stack then the natural map Pic(X)Q → A1op (X)Q , L 7→ c1 (L) is an isomorphism.
1. Introduction A long standing problem is to extend the classical intersection product to singular varieties. Motivated by topology, one hopes to construct an “intersection Chow cohomology” theory analogous to Goresky and MacPherson’s intersection homology. There have been various approaches from this point of view using motivic theories, for example [CH1], [CH2], [FR]. Earlier, Fulton and MacPherson [FM] defined a formal Chow cohomology ring, which we call the operational Chow ring, and proved that it equals the classical intersection ring for smooth varieties. While the operational Chow ring enjoys many natural functorial properties, the product structure, which is given by composition of operations, does not have a natural interpretation in terms of intersecting subvarieties. In this paper we focus on a class of varieties which we call reductive quotient varieties. These are varieties (or more generally algebraic spaces) which are ´etale locally good quotients of smooth varieties by reductive groups. When a reductive group acts properly the quotient variety has finite quotient singularities, and there is a well-developed geometric intersection theory on such varieties [Vis], [EG]. DOI: 10.1007/S00031-020-09553-9 by Simons Collaboration Grant 315460. Supported by a Discovery Grant from the National Science and Engineering Board of Canada. Received August 7, 2018. Accepted January 24, 2019. Corresponding Author: Dan Edidin, e-mail: [email protected] ∗ Supported ∗∗
DAN EDIDIN, MATTHEW SATRIANO
However, when the stabilizers are positive-dimensional the singularities are worse and there is no intersection ring. For example, the cone over the quadric surface is the good quotient of A4 by Gm . More generally, the Cox construction [Cox] shows that all normal toric varieties are good quotients of open sets in affine space. However, only simplicial toric varieties have finite quotient singularities. p If X = Z/G is a quotient variety with Z smooth, then the quotient map Z → X induces a
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