Kieran G O’Grady 
Moduli of Double EPW-Sextics [PDF ebook] 

The author studies the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of $/bigwedge^3{/mathbb C}^6$ modulo the natural action of $/mathrm{SL}_6$, call it $/mathfrak{M}$. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK $4$-folds of Type $K3^{[2]}$ polarized by a divisor of square $2$ for the Beauville-Bogomolov quadratic form. The author will determine the stable points. His work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic $4$-folds.