Twisted Fourier–Mukai transforms for holomorphic symplectic four-folds

We apply the methods of C a˘ldăraru to construct a twisted Fourier–Mukai transform between a pair of holomorphic symplectic four-folds which are fibred by Lagrangian abelian surfaces. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain Lagrangian fibration and the derived category of twisted sheaves on its ‘mirror’ partner. As a corollary, we extend the original Fourier–Mukai transform to degenerations of abelian surfaces. Another consequence of the general theory is that the holomorphic symplectic four-fold and its mirror are connected by a one-parameter family of deformations through Lagrangian fibrations.