On symplectic birational self-maps of projective hyperk\"{a}hler manifolds of K3$^{[n]}$-type

TL;DR

This paper characterizes projective hyperk"{a}hler} manifolds of K3$^{[n]}$-type with symplectic birational self-maps, showing they are moduli spaces of stable sheaves on K3 surfaces, and explores their birational involutions.

Contribution

It establishes a link between symplectic birational self-maps and moduli spaces of sheaves on K3 surfaces, providing new insights into their structure and symmetries.

Findings

Manifolds with non-trivial symplectic birational self-maps are moduli spaces of sheaves.

Reflections on the movable cone can originate from birational involutions.

Characterization of when reflections correspond to involutions.

Abstract

We prove that projective hyperk\"{a}hler manifolds of K3$^{[n]}$-type admitting a non-trivial symplectic birational self-map of finite order are isomorphic to moduli spaces of stable (twisted) coherent sheaves on K3 surfaces. Motivated by this result, we analyze the reflections on the movable cone of moduli spaces of sheaves and determine when they come from a birational involution.