Uniqueness of crepant resolutions and symplectic singularities
Annales de l'Institut Fourier, Volume 54 (2004) no. 1, pp. 1-19.

We prove the uniqueness of crepant resolutions for some quotient singularities and for some nilpotent orbits. The finiteness of non-isomorphic symplectic resolutions for 4- dimensional symplectic singularities is proved. We also give an example of a symplectic singularity which admits two non-equivalent symplectic resolutions.

Nous démontrons l'unicité des résolutions crépantes pour certaines singularités quotient et pour certaines adhérences d'orbites nilpotentes. La finitude des résolutions symplectiques non-isomorphes pour les singularités symplectiques de dimension 4 est démontrée. Nous construisons aussi un exemple d'une singularité symplectique qui admet deux résolutions symplectiques non-équivalentes.

DOI: 10.5802/aif.2008
Classification: 14E15
Keywords: crepant resolutions, symplectic singularities
Mot clés : résolutions crépantes, singularités symplectiques
Fu, Baohua 1; Namikawa, Yoshinori 2

1 Université de Nice, Parc Valrose, Laboratoire J.A. Dieudonné, 06108 Nice cedex 02 (France)
2 Departement of Mathematics, Kyoto University, Graduate School of Science, Kiat-Shirakawa Oiwake-cho, Kyoto 606-8502 (Japon)
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Fu, Baohua; Namikawa, Yoshinori. Uniqueness of crepant resolutions and symplectic singularities. Annales de l'Institut Fourier, Volume 54 (2004) no. 1, pp. 1-19. doi : 10.5802/aif.2008. https://aif.centre-mersenne.org/articles/10.5802/aif.2008/

[Bea] A. Beauville Symplectic singularities, Invent. Math, Volume 139 (2000), pp. 541-549 | MR | Zbl

[CMS] K. Cho; Y. Miyaoka; N.I. Shepherd-Barron Characterizations of projective space and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997) (Adv. Stud. Pure Math), Volume 35 (2002), pp. 1-88 | MR | Zbl

[Deb] O. Debarre Higher-Dimensional Algebraic Geometry, Universitext, Springer Verlag, 2001 | MR | Zbl

[Fu1] B. Fu Symplectic resolutions for nilpotent orbits, Invent. Math, Volume 151 (2003), pp. 167-186 | MR | Zbl

[Fu2] B. Fu Symplectic resolutions for nilpotent orbits (II), C. R. Math. Acad. Sci. Paris, Volume 336 (2003), pp. 277-281 | MR | Zbl

[Fuj] A. Fujiki On primitively symplectic compact Kähler V-manifolds of dimension four, Classification of algebraic and analytic manifolds (Katata, 1982) (Progr. Math.), Volume 39 (1983), pp. 71-250 | MR | Zbl

[Got] R. Goto On toric hyper-Kähler manifolds given by the hyper-Kähler quotient method, Infinite analysis, Part A, B (Kyoto, 1991) (Adv. Ser. Math. Phys.), Volume 16 (1991), pp. 317-338 | MR | Zbl

[Hes] W.H. Hesselink Polarizations in the classical groups, Math. Z, Volume 160 (1978), pp. 217-234 | EuDML | MR | Zbl

[Huy] D. Huybrechts Compact hyper-Kähler manifolds: basic results, Invent. Math, Volume 135 (1999), pp. 63-113 | MR | Zbl

[Ka1] D. Kaledin Dynkin diagrams and crepant resolutions of quotient singularities (e-print. To appear in Selecta Math, math.AG/9903157)

[Ka2] D. Kaledin McKay correspondence for symplectic quotient singularities, Invent. Math, Volume 148 (2002), pp. 151-175 | MR | Zbl

[Ka3] D. Kaledin Symplectic resolutions: deformations and birational maps (e-print, math.AG/0012008)

[KM] Y. Kawamata; K. Matsuki The number of the minimal models for a 3-fold of general type is finite, Math. Ann., Volume 276 (1987), pp. 595-598 | MR | Zbl

[KP] H. Kraft; C. Procesi On the geometry of conjugacy classes in classical groups, Comment. Math. Helv, Volume 57 (1982), pp. 539-602 | MR | Zbl

[Mat] K. Matsuki Termination of flops for 4-folds, Amer. J. Math, Volume 113 (1991), pp. 835-859 | MR | Zbl

[Na1] Y. Namikawa Deformation theory of singular symplectic n-folds, Math. Ann, Volume 319 (2001), pp. 597-623 | MR | Zbl

[Na2] Y. Namikawa Mukai flops and derived categories II (e-print, math.AG/0305086) | MR

[Sho] V.V. Shokurov Prelimiting flips, Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry (Tr. Mat. Inst. Steklova), Volume 240 (2003), pp. 82-219 | MR | Zbl

[Wi1] J. Wierzba Contractions of symplectic varieties, J. Algebraic Geom, Volume 12 (2003), pp. 507-534 | MR | Zbl

[Wi2] J. Wierzba Symplectic Singularities (2000) (Ph. D. thesis, Trinity College, Cambridge University)

[WW] J. Wierzba; J.A. Wisniewski Small contractions of symplectic 4-folds, Duke Math. J., Volume 120 (2003) no. math. AG/0201028, pp. 65-95 | MR | Zbl

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