Deformations of singular symplectic varieties and termination of the log minimal model program

We generalize Huybrechts' theorem on deformation equivalence of birational irreducible symplectic manifolds to the singular setting. More precisely, under suitable natural hypotheses, we show that two birational symplectic varieties are locally trivial deformations of each other. As an application we show the termination of any log minimal model program for a pair (X; Delta) of a projective irreducible symplectic manifold X and an effective R-divisor Delta. To prove this result we follow Shokurov's strategy and show that LSC and ACC for minimal log discrepancies hold for all the models appearing along any log MMP of the initial pair.