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- Title
Symplectic resolutions for nilpotent orbits.
- Authors
Fu, Baohua
- 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:Z⇾𝒪tilde;, 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.
- Subjects
FINITE groups; NILPOTENT groups; NILPOTENT Lie groups; MATHEMATICS; MATHEMATICAL analysis; ALGEBRA
- Publication
Inventiones Mathematicae, 2003, Vol 151, Issue 1, p167
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-002-0260-9