Fourier-Mukai partners of a K3 surface and the cusps of its Kahler moduli

TL;DR

This paper establishes a correspondence between Fourier-Mukai partners of a projective K3 surface and the cusps of its Kahler moduli space, using lattice theory to connect geometric and arithmetic structures.

Contribution

It introduces a novel lattice-theoretic framework linking Fourier-Mukai partners with the cusps of the Kahler moduli space of K3 surfaces.

Findings

One-to-one correspondence between Fourier-Mukai partners and 0-dimensional cusps.

Relation between twisted Fourier-Mukai partners and general 0-dimensional cusps.

Connection between elliptic fibrations and 1-dimensional cusps.

Abstract

Using lattice theory, we establish a one-to-one correspondence between the set of Fourier-Mukai partners of a projective $K3$ surface and the set of 0-dimensional standard cusps of its Kahler moduli. We also study the relation between twisted Fourier-Mukai partners and general 0-dimensional cusps, and the relation between Fourier-Mukai partners with elliptic fibrations and certain 1-dimensional cusps.