Symplectic resolutions for nilpotent orbits

By Baohua Fu. 2002

102 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/18aa7d1a392fa686b523f5530373b05158450262

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In Geometry and Topology

1. Oscar Chacaltana
2. J. Distler
3. Paul Blain Levy
4. Y. Namikawa
5. Baohua Fu
6. Yuji Tachikawa
7. Ben Webster
8. D. Juteau

In Mathematical Physics

1. Yuji Tachikawa
2. Oscar Chacaltana
3. J. Distler
4. Paul Blain Levy
5. Y. Namikawa
6. Baohua Fu
7. Ben Webster
8. I. Losev

Citations in the corpus

• Generic singularities of nilpotent orbit closures (2015, background): “By [18], those are the only symplectic resolutions of nilpotent orbit closures.”
• Quantizations of conical symplectic resolutions I: local and global structure (2012, background): “symbols Mand Xdenote smooth varieties. Not every symplectic variety (in the sense of Beauville [Bea00]) admits a symplectic resolution; for example, closures of non-Richardson nilpotent orbits do not [Fu03]. On the other hand, every symplectic variety has a crepant partial resolution Mwhich is termi”
• La singularité de O’Grady (2005, unknown): “Il est connue que Z est l’adhérence de l’orbite nilpotente de type o(2, 2) (voir [5], [8] pour plus de détails).”
• Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories (2012, background): “arities of a finite cover of the orbit Ox = Ind g l(O l e). When Ol is the zero orbit, X 0 is just T∗(GC/PC) and is smooth, which can then be blown down to the closure of a finite cover of Ox. See also [48, 49]. With this procedure, the induction from lgives O [3,3,1,1], while the induction from l ′ g”
• Poisson deformations of affine symplectic varieties (2006, background): “losure O¯ of O in g. Then 1 LOCALSYSTEMASSOCIATEDWITHASYMPLECTICVARIETY4 O˜ is an affine symplectic variety with the Kostant-Kirillov 2-form ωon O. Let Gbe a complex algebraic group with Lie(G) = g. By [Fu], O˜ has a crepant projective resolution if and only if Ois a Richardson orbit (cf. [CM]) and th”