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

第二上同调的周期决定了复不可约辛流形的同构类吗？这就是托雷利问题。对于K3曲面是正确的，但是对于dim（大于或等于）4存在Debarre的反例。Mukai， Huybrechts和其他一些人提出了修改原始托雷利问题的双民族托雷利问题。我们甚至为这个问题构造了一个反例。另一方面，对于一个复杂的不可约辛流形，从相干束的派生范畴中可以恢复出原始流形的哪一类信息？什么时候我们有两个复不可约流形的派生范畴的等价？这样的问题很有趣。证明了两个光滑射影变数由一个Mukai翻牌连接时所导出的范畴是等价的。作为一种应用，我们知道双等价的、复的、射影辛四折具有等价的派生范畴。研究了复辛变的变形理论，并将复辛流形的几个重要事实推广到奇异辛变。

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.