Cohomology of the moduli space of cubic threefolds and its smooth models

TL;DR

This paper computes and compares the intersection cohomology of different compactifications of the moduli space of cubic threefolds, revealing their topological relationships and boundary behaviors.

Contribution

It provides a detailed cohomological analysis of various geometric compactifications of the cubic threefold moduli space using Kirwan's method and the decomposition theorem.

Findings

Cohomology groups of GIT and ball quotient compactifications are explicitly computed.

The behavior of cohomology under birational maps is characterized.

Boundary structures of the ball quotient model are thoroughly analyzed.

Abstract

We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily-Borel and toroidal compactifications of the ball quotient model, due to Allcock-Carlson-Toledo. Our starting point is Kirwan's method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli of cubic surfaces is discussed in an appendix.