Rank Two Fourier-Mukai Transforms for K3 Surfaces

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 XX and YY with negative twisted determinant is that YY is isomorphic to XX 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.