  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.
