Finite order symplectic birational self-maps on Kummer-type manifolds

TL;DR

This paper characterizes finite-order symplectic birational self-maps on Kummer-type hyperk"ahler manifolds, linking them to twisted modular structures and analyzing their cohomological actions.

Contribution

It provides a classification of such self-maps, identifies exceptions based on Néron-Severi lattices, and explores their relation to moduli spaces of twisted sheaves.

Findings

Most Kummer-type manifolds with certain symplectic self-maps are twisted modular.

Complete characterization of exceptions via Néron-Severi lattices.

Determined conditions under which birational transformations correspond to finite-order symplectic maps.

Abstract

A projective hyperk\"ahler manifold of Kummer-type is said to be twisted modular if it is birational to the Albanese fiber of a moduli space of twisted sheaves on an abelian surface. We prove that, with the exception of certain cases of Picard rank 3, any projective Kummer-type manifold admitting a finite-order symplectic birational self-map that acts nontrivially on its second cohomology group is twisted modular. We provide a complete characterization of these exceptions in terms of their N\'eron-Severi lattices. We then investigate symplectic birational self-maps of modular Kummer-type manifolds, determining exactly which Mukai vectors allow the birational transformation induced by crossing the vertical wall, which acts on cohomology as a reflection, to correspond to a finite-order symplectic birational self-map. Additionally, we prove in an appendix several results concerning moduli spaces of twisted sheaves on abelian surfaces which were not readily available in the literature.