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 an 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.
Subject
Algebra and Number Theory
Reference55 articles.
1. Footnotes to papers of O’Grady and Markman
2. Every rational Hodge isometry between two K3K3 surfaces is algebraic
3. The weight-two Hodge structure of moduli spaces of sheaves on a $K3$ surface;O'Grady;J. Algebraic Geom,1997