Let $M$ be a K3 surface or a 2-dimensional compact torus, $B$ a stable bundle on $M$, and $X$ the coarse moduli of stable deformations of $B$. The Fourier-Mukai transform is a functor $FM:\; D_b(M) \arrow D_b(X)$, where $D_b(M)$, $D_b(X)$ denotes the derived category of coherent sheaves on $M$, $X$.
Consider a stable bundle $B_1$ on $M$, and let $FM(B_1)$ be the corresponding complex of coherent sheaves on $X$. Assume $X$ is compact. It was conjectured that the cohomology of the complex $FM(B_1)$ are polystable sheaves on $X$. This conjecture is known when the bundles $B$, $B_1$ have zero degree. We prove it for arbitrary stable bundles $B$, $B_1$.
