Moduli of double EPW-sextics

We study the GIT quotient of the set of lagrangian subspaces of the third wedge-product of a 6-dimensional complex vector space, modulo the natural action of SL_6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of hyperkaehler 4-folds deformation equivalent to the Hilbert square of a K3, polarized by a divisor of square 2 for the Beauville-Bogomolov quadratic form. We determine the stable points. Our work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic 4-folds. We prove a result which is analogous to a theorem of Laza asserting that cubic 4-folds with simple singularities are stable. We describe the irreducible components of the GIT boundary of M. Our final goal (not achieved in this work) is to understand completely the period map from M to the Baily-Borel compactification of the relevant period domain modulo an arithmetic group. We will analyze the locus in the GIT-boundary of M where the period map is not regular. Our results suggest that M is isomorphic to Looijenga’s compactification associated to 3 specific hyperplanes in the period domain.