Moduli of double EPW-sextics

We will study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of






⋀


3




C


6


\bigwedge ^3{\mathbb C}^6



modulo the natural action of SL





6

_6



, call it




M

\mathfrak {M}



. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK



4
4



-folds of Type




K

3

[
2
]



K3^{[2]}



polarized by a divisor of square



2
2



for the Beauville-Bogomolov quadratic form. We will 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
4



-folds. We will prove a result which is analogous to a theorem of Laza asserting that cubic



4
4



-folds with simple singularities are stable. We will also describe the irreducible components of the GIT boundary of




M

\mathfrak {M}



. Our final goal (not achieved in this work) is to understand completely the period map from




M

\mathfrak {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

\mathfrak {M}



where the period map is not regular. Our results suggest that




M

\mathfrak {M}



is isomorphic to Looijenga’s compactification associated to



3
3



specific hyperplanes in the period domain.