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In this paper, firstly we calculate Picard groups of a nilpotent orbit O in a classical complex simple Lie algebra and discuss the properties of being Q-factorial and factorial for the normalization Õ of the closure of O. Then we consider the problem of symplectic resolutions for Õ. Our main theorem says that for any nilpotent orbit O in a semi-simple… (More)
We give two characterizations of hyperquadrics: one as non-degenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as LQEL-manifolds with large secant defects.
Based on our previous work , we prove that for any two projective symplectic resolutions Z1 and Z2 for a nilpotent orbit closure in a simple complex Lie algebra of classical type, then Z1 is deformation equivalent to Z2. This provides support for a “folklore” conjecture on symplectic resolutions for symplectic singularities.
We prove the uniqueness of crepant resolutions for some quotient singularities and for some nilpotent orbits. The finiteness of nonisomorphic symplectic resolutions for 4-dimensional symplectic singularities is proved. We also give an example of symplectic singularity which admits two non-equivalent symplectic resolutions.
Let X be a smooth irreducible complex variety and G a finite subgroup of Aut(X). There is a natural action of G on T ∗X preserving the canonical symplectic form. We show that if T ∗X/G admits a symplectic resolution π : Z → T ∗X/G, then X/G is smooth and Z contains an open set isomorphic to T ∗(X/G). In the case of X = P and G ⊂ SL(n + 1,C), we give an… (More)
Let O be a nilpotent orbit in a semisimple complex Lie algebra g. Denote by G the simply connected Lie group with Lie algebra g. For a G-homogeneous covering M → O, let X be the normalization of O in the function field of M . In this note, we study the existence of symplectic resolutions for such coverings X.
For stratified Mukai flops of type An,k,D2k+1 and E6,I , it is shown the fiber product induces isomorphisms on Chow motives. In contrast to (standard) Mukai flops, the cup product is generally not preserved. For An,2, D5 and E6,I flops, quantum corrections are found through degeneration/deformation to ordinary flops.
Let S be a smooth complex connected analytic surface which admits a holomorphic symplectic structure. Let S(n) be its nth symmetric product. We prove that every projective symplectic resolution of S(n) is isomorphic to the Douady-Barlet resolution S[n] → S(n).