Cohomology of moduli space of cubic fourfolds I

TL;DR

This paper computes the cohomology and Betti numbers of the moduli space of cubic fourfolds with ADE singularities, using Kirwan's blowup and Laza's GIT construction, and extends results to the Baily-Borel compactification.

Contribution

It provides explicit Betti numbers for the moduli space and its compactification, advancing understanding of their topological structure.

Findings

Betti numbers of Kirwan's resolution are obtained.

Betti numbers of the intersection cohomology of the compactification are computed.

The cohomology of the moduli space with ADE singularities is explicitly described.

Abstract

In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan's blowup and Laza's GIT construction. More precisely, we obtain the Betti numbers of Kirwan's resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.