Moduli of Double EPW-Sextics

by Kieran G. O’Grady
The author studies the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of ⋀3C6 modulo the natural action of SL6, call it 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.

Abstract

We will study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of ⋀3C6 modulo the natural action of \SL6, call it 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. 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-folds. We will prove a result which is analogous to a theorem of Laza asserting that cubic 4-folds with simple singularities are stable. We will also 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.

Table of Contents

Introduction
Preliminaries
One-parameter subgroups and stability
Plane sextics and stability of lagrangians
Lagrangians with large stabilizers
Description of the GIT-boundary
Boundary components meeting I in a subset of XW∪{x,x∨}
The remaining boundary components
Appendix A. Elementary auxiliary results
Appendix B. Tables
Bibliography

More info and Hardcover at AMS

Donate to Support :)