Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic 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.

Var3: The paper is reorganized, it starts with description of self-correspondences of K3 for arbitrary Picard number (the main result), and finishes with their applications to K3 with reflective Picard lattices

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