Abstract.
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 𝒪tilde; of the closure of 𝒪. Then we consider the problem of symplectic resolutions for 𝒪tilde;. Our main theorem says that for any nilpotent orbit 𝒪 in a semi-simple complex Lie algebra, equipped with the Kostant-Kirillov symplectic form ω, if for a resolution π:Z𝒪tilde;, the 2-form π*(ω) defined on π−1(𝒪) extends to a symplectic 2-form on Z, then Z is isomorphic to the cotangent bundle T *(G/P) of a projective homogeneous space, and π is the collapsing of the zero section. It proves a conjecture of Cho-Miyaoka-Shepherd-Barron in this special case. Using this theorem, we determine all varieties 𝒪tilde; which admit such a resolution.
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Oblatum 6-V-2002 & 7-VIII-2002¶Published online: 10 October 2002
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Fu, B. Symplectic resolutions for nilpotent orbits. Invent. math. 151, 167–186 (2003). https://doi.org/10.1007/s00222-002-0260-9
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DOI: https://doi.org/10.1007/s00222-002-0260-9