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Moduli of double EPW-sextics


About this Title

Kieran G. O’Grady

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 240, Number 1136
ISBNs: 978-1-4704-1696-6 (print); 978-1-4704-2824-2 (online)
DOI: http://dx.doi.org/10.1090/memo/1136
Published electronically: October 9, 2015
Keywords:GIT quotient, period map, hyperkähler varieties

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. One-parameter subgroups and stability
  • Chapter 4. Plane sextics and stability of lagrangians
  • Chapter 5. Lagrangians with large stabilizers
  • Chapter 6. Description of the GIT-boundary
  • Chapter 7. Boundary components meeting $\mathfrak I$ in a subset of $\mathfrak X_\mathcal W\cup \{\mathfrak x,\mathfrak x^\vee \}$
  • Chapter 8. The remaining boundary components
  • Appendix A. Elementary auxiliary results
  • Appendix B. Tables

Abstract


We will study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of $⋀^{3}\mathbb C^{6}$ modulo the natural action of 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. 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 $\mathfrak M$. Our final goal (not achieved in this work) is to understand completely the period map from $\mathfrak M$ to the Baily-Borel compactification of the relevant period domain modulo an arithmetic group. We will analyze the locus in the GIT-boundary of $\mathfrak M$ where the period map is not regular. Our results suggest that $\mathfrak M$ is isomorphic to Looijenga’s compactification associated to $3$ specific hyperplanes in the period domain.

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