Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups

TL;DR

This paper characterizes K3 surfaces with reflective Picard lattices through their self-correspondences via moduli of sheaves, linking geometric properties with arithmetic hyperbolic reflection groups and revealing their finite generative structure.

Contribution

It introduces a novel characterization of reflective K3 surfaces using self-correspondences and connects geometric moduli spaces with arithmetic reflection groups.

Findings

K3 surfaces with reflective Picard lattices are characterized by compositions of simple self-correspondences.

Self-correspondences with integral action are equivalent to finite compositions of basic moduli-based correspondences.

The work links geometric properties of K3 surfaces to the structure of hyperbolic reflection groups.

Abstract

An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-correspondences via moduli of sheaves with primitive isotropic Mukai vector: Their self-correspondences with integral action on the Picard lattice are numerically equivalent to compositions of a finite number of especially simple self-correspondences via moduli of sheaves. This relates two topics: Self-correspondences of K3 surfaces via moduli of sheaves and Arithmetic hyperbolic reflection groups. It also raises several natural unsolved related problems.