A survey on symplectic singularities and symplectic resolutions
Annales mathématiques Blaise Pascal, Tome 13 (2006) no. 2, pp. 209-236.

This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions.

DOI : 10.5802/ambp.218
Baohua Fu 1

1 Laboratoire J. Leray Université de Nantes, Faculté des sciences 2, Rue de la Houssinière BP 92208, F-44322 Nantes Cedex 03 France
@article{AMBP_2006__13_2_209_0,
     author = {Baohua Fu},
     title = {A survey on symplectic singularities and symplectic resolutions},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {209--236},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {13},
     number = {2},
     year = {2006},
     doi = {10.5802/ambp.218},
     mrnumber = {2275448},
     zbl = {1116.14008},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.218/}
}
TY  - JOUR
AU  - Baohua Fu
TI  - A survey on symplectic singularities and symplectic resolutions
JO  - Annales mathématiques Blaise Pascal
PY  - 2006
SP  - 209
EP  - 236
VL  - 13
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.218/
DO  - 10.5802/ambp.218
LA  - en
ID  - AMBP_2006__13_2_209_0
ER  - 
%0 Journal Article
%A Baohua Fu
%T A survey on symplectic singularities and symplectic resolutions
%J Annales mathématiques Blaise Pascal
%D 2006
%P 209-236
%V 13
%N 2
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.218/
%R 10.5802/ambp.218
%G en
%F AMBP_2006__13_2_209_0
Baohua Fu. A survey on symplectic singularities and symplectic resolutions. Annales mathématiques Blaise Pascal, Tome 13 (2006) no. 2, pp. 209-236. doi : 10.5802/ambp.218. https://ambp.centre-mersenne.org/articles/10.5802/ambp.218/

[1] V. Batyrev Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), Publish or Perish, Inc., Houston, 1998, pp. 1-32 | MR

[2] V. Batyrev Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc., Volume 1 (1999), pp. 5-33 | DOI | MR | Zbl

[3] A. Beauville Fano contact manifolds and nilpotent orbits, Comment. Math. Helv, Volume 73 (1998), pp. 566-583 | DOI | MR | Zbl

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

[5] R. Bezrukavnikov; D. Kaledin McKay equivalence for symplectic resolutions of singularities, Proc. Steklov Inst. Math., Volume 246 (2004), pp. 13-33 | MR | Zbl

[6] A. Bialynicki-Birula Some theorems on actions of algebraic groups, Ann. of Math. (2), Volume 98 (1973), pp. 480-497 | DOI | MR | Zbl

[7] F. Bottacin Poisson structures on moduli spaces of sheaves over Poisson surfaces, Invent. Math., Volume 121 (1995), pp. 421-436 | DOI | MR | Zbl

[8] D. Burns; Y. Hu; T. Luo HyperKähler manifolds and birational transformations in dimension 4, Vector bundles and representation theory (Columbia, MO, 2002), Amer. Math. Soc., 2003, pp. 141-149 | MR | Zbl

[9] Y. Cho; Y. Miyaoka; N. Shepherd-Barron; Miyaoka; Mori Characterizations of projective space and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997), Math. Soc. Japan, 2002, pp. 1-88 | MR | Zbl

[10] J. Choy; Y.-H. Kiem Nonexistence of crepant resolution of some moduli spaces of sheaves on a K3 surface (2004) (math.AG/0407100)

[11] J. Choy; Y.-H. Kiem On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface, Math. Z., Volume 252 (2006), pp. 557-575 | DOI | MR

[12] A. M. Cohen Finite quaternionic reflection groups, J. Algebra, Volume 64 (1980), pp. 293-324 | DOI | MR | Zbl

[13] D. Collingwood; W. Mc Govern Nilpotent orbits in semi-simple Lie algebras, Van Nostrand Reinhold Co., New York, 1993 | MR | Zbl

[14] S. Druel Singularités symplectiques, J. Algebraic Geom., Volume 13 (2004), pp. 427-439 | MR | Zbl

[15] B. Fu Symplectic resolutions for coverings of nilpotent orbits, C. R. Acad. Sci., Volume 336 (2003), pp. 159-162 | MR | Zbl

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

[17] B. Fu Symplectic resolutions for nilpotent orbits (II), C. R. Acad. Sci., Volume 337 (2003), pp. 277-281 | MR | Zbl

[18] B. Fu Birational geometry in codimension 2 of symplectic resolutions (2004) (math.AG/0409224)

[19] B. Fu Extremal contractions, stratified Mukai flops and Springer maps (2006) (math.AG/0605431)

[20] B. Fu Mukai flops and deformations of symplectic resolutions, Math. Z., Volume 253 (2006), pp. 87-96 | DOI | MR | Zbl

[21] B. Fu; Y. Namikawa Uniqueness of crepant resolutions and symplectic singularities, Ann. Inst. Fourier, Volume 54 (2004), pp. 1-19 | DOI | Numdam | MR | Zbl

[22] V. Ginzburg; D. Kaledin Poisson deformations of symplectic quotient singularities, Adv. Math., Volume 186 (2004), pp. 1-57 | DOI | MR | Zbl

[23] I. Gordon Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc., Volume 35 (2003), pp. 321-336 | DOI | MR | Zbl

[24] Robert M. Guralnick; J. Saxl Generation of finite almost simple groups by conjugates, J. Algebra, Volume 268 (2003), pp. 519-571 | DOI | MR | Zbl

[25] R. Hartshorne Algebraic Geometry, Springer-Verlag, 1977 | MR | Zbl

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

[27] Y. Hu Geometric Invariant Theory and Birational Geometry (2005) (math.AG/0502462)

[28] Y. Hu; S.-T. Yau HyperKähler manifolds and birational transformations, Adv. Theor. Math. Phys., Volume 6 (2002), pp. 557-574 | MR | Zbl

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

[30] D. Kaledin Symplectic singularities from the Poisson point of view, J. Reine Angew. Math.

[31] D. Kaledin Symplectic resolutions: deformations and birational maps (2000) (math.AG/0012008)

[32] D. Kaledin McKay correspondence for symplectic quotient singularities, Invent. math., Volume 148 (2002), pp. 150-175 | DOI | MR | Zbl

[33] D. Kaledin On crepant resolutions of symplectic quotient singularities, Selecta Math. (N.S.), Volume 9 (2003), pp. 529-555 | DOI | MR | Zbl

[34] D. Kaledin Derived equivalence by quantization (2005) (math.AG/0504584)

[35] D. Kaledin; M. Lehn Local structure of hyperKaehler singularities in O’Grady’s examples (2004) (math.AG/0405575)

[36] D. Kaledin; M. Lehn; C. Sorger Singular symplectic moduli spaces, Invent. Math., Volume 164 (2006), pp. 591-614 | DOI | MR | Zbl

[37] Y. Kawamata D-equivalence and K-equivalence, J. Differential Geom., Volume 61 (2002), pp. 147-171 | MR | Zbl

[38] H. Kraft; C. Procesi Closures of conjugacy classes of matrices are normal, Invent. Math., Volume 53 (1979), pp. 227-247 | DOI | MR | Zbl

[39] E. Markman Brill-Noether duality for moduli spaces of sheaves of K3 surfaces, J. Algebr. Geom., Volume 10 (2001), pp. 623-694 | MR | Zbl

[40] S. Mukai Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math., Volume 77 (1984), pp. 101-116 | DOI | MR | Zbl

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

[42] Y. Namikawa Extension of 2-forms and symplectic varieties, J. Reine Angew. Math., Volume 539 (2001), pp. 123-147 | DOI | MR | Zbl

[43] Y. Namikawa A note on symplectic singularitie (2001) (math.AG/0101028)

[44] Y. Namikawa Birational geometry of symplectic resolutions of nilpotent orbits (2004) (math.AG/0404072)

[45] Y. Namikawa Birational geometry of symplectic resolutions of nilpotent orbits II (2004) (math.AG/0408274)

[46] Y. Namikawa Flops and Poisson deformations of symplectic varieties (2005) (math.AG/0510059)

[47] Y. Namikawa On deformations of Q-factorial symplectic varieties (2005) (math.AG/0506534)

[48] K. O’Grady Desingularized moduli spaces of sheaves on a K3, J. reine angew. Math., Volume 512 (1999), pp. 49-117 | DOI | MR | Zbl

[49] K. O’Grady A new six-dimensional irreducible symplectic variety, J. Algebraic Geom., Volume 12 (2003), pp. 435-505 | DOI | MR | Zbl

[50] D. Panyushev Rationality of singularities and the Gorenstein property for nilpotent orbits, Funct. Anal. Appl., Volume 25 (1991), pp. 225-226 | DOI | MR | Zbl

[51] M. Verbitsky Holomorphic symplectic geometry and orbifold singularities, Asian J. Math., Volume 4 (2000), pp. 553-563 | MR | Zbl

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

[53] J. Wierzba; J. A. Wisniewski Small contractions of symplectic 4-folds, Duke Math. J., Volume 120 (2003), pp. 65-95 | DOI

Cité par Sources :