Symplectic resolutions for nilpotent orbits |
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Authors: | Baohua Fu |
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Institution: | (1) Université de Nice Sophia-Antipolis, Laboratoire J.A. Dieudonné, UMR CNRS 6621, Parc Valrose, 06108 Nice Cedex 2, France (e-mail: baohua.fu@polytechnique.org), FR |
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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.
Oblatum 6-V-2002 & 7-VIII-2002?Published online: 10 October 2002 |
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