Sixfolds of generalized Kummer type and K3 surfaces

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.

arXiv commentv3: final version, to appear in Compositio Mathematica

References in corpus (7)

• The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface
• The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces
• On two rationality conjectures for cubic fourfolds
• Induced Automorphisms on Irreducible Symplectic Manifolds
• The Hodge diamond of O'Grady's 6-dimensional example
• Monodromy Invariants and Polarization Types of Generalized Kummer Fibrations
• Symplectic involutions of $K{3}^{[n]}$ type and Kummer $n$ type manifolds

Cited by in corpus (1)

• Algebraic cycles on hyper-Kähler varieties of generalized Kummer type