Equivariant geometry and the cohomology of the moduli space of curves

Abstract

In this expository article we give a categorical definition of the integral cohomology ring of a stack. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Via this identification we show that for Deligne-Mumford quotient stacks this cohomology is rationally isomorphic to the rational cohomology of the coarse moduli space. The theory is presented with a focus on the stacks of smooth and stable curves.

arXiv comment34 pages - to appear in forthcoming Handbook of Moduli, edited by G. Farkas and I. Morrison

References in corpus (2)

• The Stack of Rational Nodal Curves
• The Chow ring of the stack of cyclic covers of the projective line

Cited by in corpus (10)

• Tropical curves, graph complexes, and top weight cohomology of M_g
• The Evaluation Space of Logarithmic Stable Maps
• The cohomology rings of smooth toric varieties and quotients of moment-angle complexes
• The Chow rings of the moduli spaces of curves of genus 7, 8, and 9
• Moment Angle Complexes and Big Cohen-Macaulayness
• Compactified universal jacobian and the double ramification cycle
• Cohomological Hall algebras and vertex algebras
• Equivariant Cohomology of Weighted Grassmannians and Weighted Schubert Classes
• Getzler-Kapranov complexes and moduli stacks of curves
• Connected sums of simplicial complexes and equivariant cohomology