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.
Keywords: crepant resolutions, symplectic singularities
Mot clés : résolutions crépantes, singularités symplectiques
Fu, Baohua 1; Namikawa, Yoshinori 2
@article{AIF_2004__54_1_1_0, author = {Fu, Baohua and Namikawa, Yoshinori}, title = {Uniqueness of crepant resolutions and symplectic singularities}, journal = {Annales de l'Institut Fourier}, pages = {1--19}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {54}, number = {1}, year = {2004}, doi = {10.5802/aif.2008}, zbl = {1063.14018}, mrnumber = {2069119}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2008/} }
TY - JOUR AU - Fu, Baohua AU - Namikawa, Yoshinori TI - Uniqueness of crepant resolutions and symplectic singularities JO - Annales de l'Institut Fourier PY - 2004 SP - 1 EP - 19 VL - 54 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2008/ DO - 10.5802/aif.2008 LA - en ID - AIF_2004__54_1_1_0 ER -
%0 Journal Article %A Fu, Baohua %A Namikawa, Yoshinori %T Uniqueness of crepant resolutions and symplectic singularities %J Annales de l'Institut Fourier %D 2004 %P 1-19 %V 54 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2008/ %R 10.5802/aif.2008 %G en %F AIF_2004__54_1_1_0
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] Symplectic singularities, Invent. Math, Volume 139 (2000), pp. 541-549 | MR | Zbl
[CMS] 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] Higher-Dimensional Algebraic Geometry, Universitext, Springer Verlag, 2001 | MR | Zbl
[Fu1] Symplectic resolutions for nilpotent orbits, Invent. Math, Volume 151 (2003), pp. 167-186 | MR | Zbl
[Fu2] Symplectic resolutions for nilpotent orbits (II), C. R. Math. Acad. Sci. Paris, Volume 336 (2003), pp. 277-281 | MR | Zbl
[Fuj] On primitively symplectic compact Kähler -manifolds of dimension four, Classification of algebraic and analytic manifolds (Katata, 1982) (Progr. Math.), Volume 39 (1983), pp. 71-250 | MR | Zbl
[Got] 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] Polarizations in the classical groups, Math. Z, Volume 160 (1978), pp. 217-234 | EuDML | MR | Zbl
[Huy] Compact hyper-Kähler manifolds: basic results, Invent. Math, Volume 135 (1999), pp. 63-113 | MR | Zbl
[Ka1] Dynkin diagrams and crepant resolutions of quotient singularities (e-print. To appear in Selecta Math, math.AG/9903157)
[Ka2] McKay correspondence for symplectic quotient singularities, Invent. Math, Volume 148 (2002), pp. 151-175 | MR | Zbl
[Ka3] Symplectic resolutions: deformations and birational maps (e-print, math.AG/0012008)
[KM] 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] On the geometry of conjugacy classes in classical groups, Comment. Math. Helv, Volume 57 (1982), pp. 539-602 | MR | Zbl
[Mat] Termination of flops for 4-folds, Amer. J. Math, Volume 113 (1991), pp. 835-859 | MR | Zbl
[Na1] Deformation theory of singular symplectic -folds, Math. Ann, Volume 319 (2001), pp. 597-623 | MR | Zbl
[Na2] Mukai flops and derived categories II (e-print, math.AG/0305086) | MR
[Sho] Prelimiting flips, Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry (Tr. Mat. Inst. Steklova), Volume 240 (2003), pp. 82-219 | MR | Zbl
[Wi1] Contractions of symplectic varieties, J. Algebraic Geom, Volume 12 (2003), pp. 507-534 | MR | Zbl
[Wi2] Symplectic Singularities (2000) (Ph. D. thesis, Trinity College, Cambridge University)
[WW] 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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