Equivariant geometry and the cohomology of the moduli space of curves
نویسنده
چکیده
In this chapter we give a categorical definition of the integral cohomology ring of a stack. For quotient stacks [X/G] the categorical cohomology ring may be identified with the equivariant cohomology H∗ G(X). Identifying the stack cohomology ring with equivariant cohomology allows us to prove that the cohomology ring of a quotient Deligne-Mumford stack is rationally isomorphic to the cohomology ring of its coarse moduli space. The theory is presented with a focus on the stacks Mg and Mg of smooth and stable curves respectively.
منابع مشابه
Cohomology of Moduli Spaces of Curves of Genus Three via Point Counts
In this article we consider the moduli space of smooth n-pointed nonhyperelliptic curves of genus three. In the pursuit of cohomological information about this space, we make Sn-equivariant counts of its numbers of points defined over finite fields for n up to seven. Together with results on the moduli spaces of smooth pointed curves of genus zero, one and two, and the moduli space of smooth hy...
متن کاملRing structures of mod p equivariant cohomology rings and ring homomorphisms between them
In this paper, we consider a class of connected oriented (with respect to Z/p) closed G-manifolds with a non-empty finite fixed point set, each of which is G-equivariantly formal, where G = Z/p and p is an odd prime. Using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a G-manifold in terms of algebra. This makes ...
متن کاملEquivariant Counts of Points of the Moduli Spaces of Pointed Hyperelliptic Curves
Abstract. In this article we consider the moduli space Hg,n of n-pointed smooth hyperelliptic curves of genus g. In order to get cohomological information we wish to make Sn-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find that there are recursion formulas in the genus that these numbers fulfill. Thus, if we can make Sn-equivariant counts of H...
متن کاملThe Moduli Space of Curves and Its Tautological Ring
The moduli space of curves has proven itself a central object in geometry. The past decade has seen substantial progress in understanding the moduli space of curves, involving ideas, for example, from geometry (algebraic, symplectic, and differential), physics, topology, and combinatorics. Many of the new ideas are related to the tautological ring of the moduli space, a subring of the cohomolog...
متن کاملChow–künneth Decomposition for Special Varieties
In this paper we investigate Murre’s conjecture on the Chow–Künneth decomposition for two classes of examples. We look at the universal families of smooth curves over spaces which dominate the moduli space Mg, in genus at most 8 and show existence of a Chow–Künneth decomposition. The second class of examples include the representation varieties of a finitely generated group with one relation. T...
متن کامل