The period map for cubic fourfolds

Understand

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.