Symplectic resolutions for nilpotent orbits

• 成果类型: Article
• 署名作者: Fu, BH
• 署名单位: Universite Cote d'Azur; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-002-0260-9
• 发表日期: 2003
• 页码: 167-186
• 关键词: deformation

摘要

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 (O) over tilde of the closure of O. Then we consider the problem of symplectic resolutions for (O) over tilde. Our main theorem says that for any nilpotent orbit O in a semi-simple complex Lie algebra, equipped with the Kostant-Kirillov symplectic form omega, if for a resolution pi : Z --> (O) over tilde the 2-form pi*(omega) defined on pi(-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 pi 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 (O) over tilde which admit such a resolution.