The period map for cubic fourfolds

TL;DR

This paper determines the image of the period map for cubic fourfolds, confirming a conjecture, and provides a new proof that this map is an open embedding, linking algebraic invariants to automorphic forms.

Contribution

It identifies the image of the period map for cubic fourfolds and relates algebraic invariants to automorphic forms, confirming a conjecture and offering a new proof of known results.

Findings

Confirmed the conjecture of Hassett on the period map image.

Provided a new proof that the period map is an open embedding.

Connected algebraic invariants with automorphic forms on a symmetric domain.

Abstract

The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding. An algebraic version of our main result is an identification of the algebra of SL(6)-invariant polynomials on the space of cubic forms in 6 complex variables with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20. We also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a Dynkin-Vinberg diagram.