Rank Two Fourier-Mukai Transforms for K3 Surfaces

TL;DR

This paper classifies rank two Fourier-Mukai transforms on K3 surfaces, revealing conditions for their existence based on determinant signs and showing all reflexive K3 surfaces admit such transforms.

Contribution

It provides a complete classification of rank two Fourier-Mukai transforms on K3 surfaces, including necessary and sufficient conditions and the existence for all reflexive K3 surfaces.

Findings

Transforms are of two types based on determinant positivity.

Existence of negative determinant transforms requires the surfaces to be isomorphic with a special line bundle.

All reflexive K3 surfaces admit Fourier-Mukai transforms.

Abstract

We study rank two locally-free Fourier-Mukai transforms on K3 surfaces and show that they come in two distinct types according to whether the determinant of a suitable twist of the kernel is positive or not. We show that a necessary and sufficient condition on the existence of Fourier-Mukai transforms of rank 2 between the derived categories of K3 surfaces X and Y with negative twisted determinant is that Y is isomorphic to X and there must exist a line bundle with no cohomology. We use these results to prove that all reflexive K3 surfaces (including the degenerate ones) admit Fourier-Mukai transforms.