Rank Two Fourier-Mukai Transforms for K3 Surfaces
arXiv:1608.04786 · doi:10.1016/j.geomphys.2017.01.020
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.
18 pages, Fixed a number of errors pointed out by the referee and also an error in the main statements kindly pointed out by Kota Yoshioka. To appear in a special edition of the Journal of Geometry and Physics in honour of Ugo Bruzzo's 60th birthday