Hodge structure on the singular cohomology of singular cubic fourfolds

Considering the singular cohomology of a cubic fourfold yields a morphism from the moduli space of (smooth) cubic fourfolds to the period domain. Both spaces have natural compactifications, the GIT moduli space of cubic fourfolds parametrizing all GIT polystable objects, and the Baily-Borel compactification of the period domain, purely coming from group theory. This yields a birational map between two projective varieties. Laza and Looijenga independently gave a precise description of the birational geometry between these two spaces. Recently, Sung Gi Park gave a systematic understanding on this picture using Hodge-Du Bois theory and higher singularities, which have been developed by Friedman-Laza, Mustata-Popa, and many others.

In order to attack this question, it is vital to understand the Hodge structure on the singular cohomology of a singular member in this moduli problem, and also the Hodge theoretic properties of a degeneration to it. In a joint work with Kenny Ascher, Jennifer Li, Lisa Marquand, Sung Gi Park, and Sasha Viktorova, we systematically carry out the calculation of the Hodge-Du Bois diamond of all GIT polystable cubic fourfolds using techniques coming from Saito’s Hodge modules.