Algebraic vaieties with Kodaira dimansion O and A generalization of the Bogomolor-decomposition

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Does the period of the second cohomology determine an isomorphism class of a complex irreducible symplectic manifold? This is the Torelli problem. It is true for a K3 surface, but there is an counter-example of Debarre for dim 【greater than or equal】 4. Mukai, Huybrechts and others have posed the birational Torelli problem modifying the original one. We construeted a counter-example even for this problem.On the other hand, for a complex irreducible symplectic manifold, which kind of information on the original variety can be recoverd from the derived category of coherent sheaves ? When do we have an equivalence of derived categories of two complex irreducible manifolds ? Such questions are quite interesting. We proved that the derived categories are equivalent if two smooth projective varieties are connected by a Mukai flop. As an application, one knows that birationally equivalent, complex, projective symplectic 4-folds have equivalent derived categories.We also studied the deformation theory of complex symplectic varieties and generalized several important facts on a complex symplectic manifold to a singular symplectic variety.