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Sixfolds of generalized Kummer type and K3 surfaces

Salvatore Floccari

Abstract

We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm{K}3^{[3]}$-type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective~$\mathrm{K}3$ surface~$S_K$. As application of this construction we show that the Kuga-Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm{K}3$ surfaces of general Picard rank $16$ satisfying the Kuga-Satake Hodge conjecture.

Sixfolds of generalized Kummer type and K3 surfaces

Abstract

We prove that any hyper-Kähler sixfold of generalized Kummer type has a naturally associated manifold of -type. It is obtained as crepant resolution of the quotient of by a group of symplectic involutions acting trivially on its second cohomology. When is projective, the variety is birational to a moduli space of stable sheaves on a uniquely determined projective~ surface~. As application of this construction we show that the Kuga-Satake correspondence is algebraic for the K3 surfaces , producing infinitely many new families of surfaces of general Picard rank satisfying the Kuga-Satake Hodge conjecture.
Paper Structure (17 sections, 24 theorems, 40 equations)

This paper contains 17 sections, 24 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Automorphisms trivial on the second cohomology
  3. Notation
  4. Relevant subvarieties
  5. Fixed loci
  6. A Lagrangian fibration
  7. Preliminaries
  8. Geometric set-up
  9. Beauville-Mukai systems
  10. The relative norm map
  11. The action of $G$
  12. The K3 surface associated to a $\mathrm{Kum}^3$-variety
  13. The singularities of $K/G$
  14. The symplectic resolution of $K/G$
  15. The associated $\mathrm{K}3$ surface
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $K$ be a manifold of $\mathrm{Kum}^3$-type. The quotient $K/G$ admits a resolution $Y_K\to K/G$ with $Y_K$ a manifold of $\mathrm{K}3^{[3]}$-type.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.4
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 46 more