Uniqueness of crepant resolutions and symplectic singularities
Annales de l'Institut Fourier, Volume 54 (2004) no. 1, p. 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 : https://doi.org/10.5802/aif.2008
Classification:  14E15
Keywords: crepant resolutions, symplectic singularities
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {1},
     year = {2004},
     pages = {1-19},
     doi = {10.5802/aif.2008},
     zbl = {1063.14018},
     mrnumber = {2069119},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_2004__54_1_1_0/

[Bea] A. Beauville Symplectic singularities, Invent. Math, Tome 139 (2000), pp. 541-549 | MR 1738060 | Zbl 0958.14001

[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), Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math) Tome 35 (2002), pp. 1-88 | MR 1929792 | Zbl 1063.14065

[Deb] O. Debarre Higher-Dimensional Algebraic Geometry, Springer Verlag, Universitext (2001) | MR 1841091 | Zbl 0978.14001

[Fu1] B. Fu Symplectic resolutions for nilpotent orbits, Invent. Math, Tome 151 (2003), pp. 167-186 | MR 1943745 | Zbl 1072.14058 | Zbl 01965461

[Fu2] B. Fu Symplectic resolutions for nilpotent orbits (II), C. R. Math. Acad. Sci. Paris, Tome 336 (2003), pp. 277-281 | MR 2009121 | Zbl 1073.14547 | Zbl 01986162

[Fuj] A. Fujiki On primitively symplectic compact Kähler V-manifolds of dimension four, Classification of algebraic and analytic manifolds (Katata, 1982), Birkhäuser, Boston (Progr. Math.) Tome 39 (1983), pp. 71-250 | MR 728609 | Zbl 0549.32018

[Got] R. Goto On toric hyper-Kähler manifolds given by the hyper-Kähler quotient method, Infinite analysis, Part A, B (Kyoto, 1991), World Sci. Publishing, River Edge, NJ (Adv. Ser. Math. Phys.) Tome 16 (1991), pp. 317-338 | MR 1187554 | Zbl 0924.53023

[Hes] W.H. Hesselink Polarizations in the classical groups, Math. Z, Tome 160 (1978), pp. 217-234 | MR 480765 | Zbl 0364.20048

[Huy] D. Huybrechts Compact hyper-Kähler manifolds: basic results, Invent. Math, Tome 135 (1999), pp. 63-113 | MR 1664696 | Zbl 0953.53031

[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, Tome 148 (2002), pp. 151-175 | MR 1892847 | Zbl 1060.14020

[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., Tome 276 (1987), pp. 595-598 | MR 879538 | Zbl 0596.14031

[KP] H. Kraft; C. Procesi On the geometry of conjugacy classes in classical groups, Comment. Math. Helv, Tome 57 (1982), pp. 539-602 | MR 694606 | Zbl 0511.14023

[Mat] K. Matsuki Termination of flops for 4-folds, Amer. J. Math, Tome 113 (1991), pp. 835-859 | MR 1129294 | Zbl 0746.14017

[Na1] Y. Namikawa Deformation theory of singular symplectic n-folds, Math. Ann, Tome 319 (2001), pp. 597-623 | MR 1819886 | Zbl 0989.53055

[Na2] Y. Namikawa Mukai flops and derived categories II (e-print, math.AG/0305086) | MR 2096144

[Sho] V.V. Shokurov Prelimiting flips, Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry (Tr. Mat. Inst. Steklova) Tome 240 (2003), pp. 82-219 | MR 1993750 | Zbl 1082.14019

[Wi1] J. Wierzba Contractions of symplectic varieties, J. Algebraic Geom, Tome 12 (2003), pp. 507-534 | MR 1966025 | Zbl 02064089

[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., Tome 120 (2003) no. math. AG/0201028, pp. 65-95 | MR 2010734 | Zbl 1036.14007