Derived and abelian equivalence of K3 surfaces
  Tom Bridgeland constructed explicit stability conditions on K3 surfaces. This paper attempts to shed more light on these particular examples, especially on the hearts of the underlying t-structures.   We prove that two K3 surfaces X and X' are derived equivalent if and only if there exist complexified polarizations B+iw and B'+iw' such that the associated abelian categories A(B+iw) and A(B'+iw') are equivalent. We study in detail the minimal objects of A(B+iw) and investigate stability under Fourier-Mukai transform. 
Daniel Huybrechts