Finite order symplectic birational self-maps on Kummer-type manifolds
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.
Cite
@article{arxiv.2605.08062,
title = {Finite order symplectic birational self-maps on Kummer-type manifolds},
author = {Yajnaseni Dutta and Dominique Mattei and Stevell Muller and Howard Nuer},
journal= {arXiv preprint arXiv:2605.08062},
year = {2026}
}