Relative \'etale slices and cohomology of moduli spaces

TL;DR

This paper establishes a local structure theorem for smooth morphisms between stacks, leading to insights into the cohomology and topology of moduli space fibers, with applications to G-bundles and sheaves.

Contribution

It introduces a new local structure theorem for smooth stacks with reductive stabilizers, revealing equisingularity and cohomological invariance of fibers.

Findings

Fibers of the moduli space have isomorphic $$-adic cohomology rings.

The family of fibers is topologically locally trivial over the base.

Intersection cohomology groups form a polarizable variation of pure Hodge structures.

Abstract

We use techniques of Alper-Hall-Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has equisingular fibers. As an application, we show that any two fibers have isomorphic $\ell$-adic cohomology rings and intersection cohomology groups. If we work over the complex numbers, we show that the family is topologically locally trivial on the base, and that the intersection cohomology groups of the fibers fit into a polarizable variation of pure Hodge structures. We apply these results to derive some consequences for the moduli spaces of $G$-bundles on smooth projective curves, and for certain moduli spaces of sheaves on del Pezzo surfaces.