We shall prove that any small deformation of a Q-factorial projectivesymplectic variety with terminal singularities is locally rigid; in otherwords, it preserves the singularity. In particular, many singular symplecticmoduli of semi-stable sheaves on K3 have no smoothings via deformations. As anapplication of the result, we also prove that the smootheness is preservedunder a flop in our symplectic case. We conjecture that a projective symplecticvariety has a smoothing by a flat deformation if and only if it has a crepant(symplectic) resolution. This conjecture would be true if the minimal modelconjecture were true.
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