Cohomological amplitude for constructible sheaves on moduli spaces of curves

TL;DR

This paper establishes bounds on the cohomology of constructible sheaves on moduli spaces of curves, recovering classical results and providing a framework for future cohomological dimension bounds.

Contribution

It introduces new bounds for the cohomology of constructible sheaves on moduli stacks of curves, extending known results and offering a method applicable to quasicoherent sheaves.

Findings

Recovered Harer's bound for the virtual cohomological dimension.

Reproduced Diaz's theorem on complete subvarieties of M_g.

Provided bounds for open subsets of the Deligne-Mumford compactification.

Abstract

We give bounds for the cohomology of constructible sheaves on the moduli stacks M_{g,n} over the complex field. This enables us recover Harer's bound for the virtual cohomological dimension of the associated mapping class groups as well the theorem of Diaz on complete subvarieties of M_g. We also obtain such bounds for any open subset of the Deligne-Mumford compactification of M_{g,n} that is a union of strata. Our proof yields a template for obtaining similar bounds for the cohomological dimension for quasicoherent sheaves on M_{g,n}.