A survey on symplectic singularities and symplectic resolutions
Fu, Baohua
Annales mathématiques Blaise Pascal, Tome 13 (2006), p. 209-236 / Harvested from Numdam

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

@article{AMBP_2006__13_2_209_0,
     author = {Fu, Baohua},
     title = {A survey on symplectic singularities and symplectic resolutions},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {13},
     year = {2006},
     pages = {209-236},
     doi = {10.5802/ambp.218},
     zbl = {1116.14008},
     mrnumber = {2275448},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2006__13_2_209_0}
}
Fu, Baohua. A survey on symplectic singularities and symplectic resolutions. Annales mathématiques Blaise Pascal, Tome 13 (2006) pp. 209-236. doi : 10.5802/ambp.218. http://gdmltest.u-ga.fr/item/AMBP_2006__13_2_209_0/

[1] Batyrev, V. 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 1672108

[2] Batyrev, V. Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc., Tome 1 (1999), pp. 5-33 | Article | MR 1677693 | Zbl 0943.14004

[3] Beauville, A. Fano contact manifolds and nilpotent orbits, Comment. Math. Helv, Tome 73 (1998), pp. 566-583 | Article | MR 1639888 | Zbl 0946.53046

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

[5] Bezrukavnikov, R.; Kaledin, D. McKay equivalence for symplectic resolutions of singularities, Proc. Steklov Inst. Math., Tome 246 (2004), pp. 13-33 | MR 2101282 | Zbl 1137.14301

[6] Bialynicki-Birula, A. Some theorems on actions of algebraic groups, Ann. of Math. (2), Tome 98 (1973), pp. 480-497 | Article | MR 366940 | Zbl 0275.14007

[7] Bottacin, F. Poisson structures on moduli spaces of sheaves over Poisson surfaces, Invent. Math., Tome 121 (1995), pp. 421-436 | Article | MR 1346215 | Zbl 0829.14019

[8] Burns, D.; Hu, Y.; Luo, T. HyperKähler manifolds and birational transformations in dimension 4, Vector bundles and representation theory (Columbia, MO, 2002), Amer. Math. Soc. (2003), pp. 141-149 | MR 1987745 | Zbl 01989688

[9] Cho, Y.; Miyaoka, Y.; Shepherd-Barron, N.; 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 1929791 | Zbl 1063.14065

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

[11] Choy, J.; Kiem, Y.-H. On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface, Math. Z., Tome 252 (2006), pp. 557-575 | Article | MR 2207759 | Zbl pre05013686

[12] Cohen, A. M. Finite quaternionic reflection groups, J. Algebra, Tome 64 (1980), pp. 293-324 | Article | MR 579063 | Zbl 0433.20035

[13] Collingwood, D.; Mc Govern, W. Nilpotent orbits in semi-simple Lie algebras, Van Nostrand Reinhold Co., New York (1993) | MR 1251060 | Zbl 0972.17008

[14] Druel, S. Singularités symplectiques, J. Algebraic Geom., Tome 13 (2004), pp. 427-439 | MR 2047675 | Zbl 1068.32018

[15] Fu, B. Symplectic resolutions for coverings of nilpotent orbits, C. R. Acad. Sci., Tome 336 (2003), pp. 159-162 | MR 1969571 | Zbl 1068.14055

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

[17] Fu, B. Symplectic resolutions for nilpotent orbits (II), C. R. Acad. Sci., Tome 337 (2003), pp. 277-281 | MR 2009121 | Zbl 1073.14547

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

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

[20] Fu, B. Mukai flops and deformations of symplectic resolutions, Math. Z., Tome 253 (2006), pp. 87-96 | Article | MR 2206638 | Zbl 1098.14009

[21] Fu, B.; Namikawa, Y. Uniqueness of crepant resolutions and symplectic singularities, Ann. Inst. Fourier, Tome 54 (2004), pp. 1-19 | Article | Numdam | MR 2069119 | Zbl 1063.14018

[22] Ginzburg, V.; Kaledin, D. Poisson deformations of symplectic quotient singularities, Adv. Math., Tome 186 (2004), pp. 1-57 | Article | MR 2065506 | Zbl 1062.53074

[23] Gordon, I. Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc., Tome 35 (2003), pp. 321-336 | Article | MR 1960942 | Zbl 1042.16017

[24] Guralnick, Robert M.; Saxl, J. Generation of finite almost simple groups by conjugates, J. Algebra, Tome 268 (2003), pp. 519-571 | Article | MR 2009321 | Zbl 1037.20016

[25] Hartshorne, R. Algebraic Geometry, Springer-Verlag (1977) | MR 463157 | Zbl 0367.14001

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

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

[28] Hu, Y.; Yau, S.-T. HyperKähler manifolds and birational transformations, Adv. Theor. Math. Phys., Tome 6 (2002), pp. 557-574 | MR 1957670 | Zbl 1044.81105

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

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

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

[32] Kaledin, D. McKay correspondence for symplectic quotient singularities, Invent. math., Tome 148 (2002), pp. 150-175 | Article | MR 1892847 | Zbl 1060.14020

[33] Kaledin, D. On crepant resolutions of symplectic quotient singularities, Selecta Math. (N.S.), Tome 9 (2003), pp. 529-555 | Article | MR 2031751 | Zbl 1066.14003

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

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

[36] Kaledin, D.; Lehn, M.; Sorger, C. Singular symplectic moduli spaces, Invent. Math., Tome 164 (2006), pp. 591-614 | Article | MR 2221132 | Zbl 1096.14037

[37] Kawamata, Y. D-equivalence and K-equivalence, J. Differential Geom., Tome 61 (2002), pp. 147-171 | MR 1949787 | Zbl 1056.14021

[38] Kraft, H.; Procesi, C. Closures of conjugacy classes of matrices are normal, Invent. Math., Tome 53 (1979), pp. 227-247 | Article | MR 549399 | Zbl 0434.14026

[39] Markman, E. Brill-Noether duality for moduli spaces of sheaves of K3 surfaces, J. Algebr. Geom., Tome 10 (2001), pp. 623-694 | MR 1838974 | Zbl 1074.14525

[40] Mukai, S. Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math., Tome 77 (1984), pp. 101-116 | Article | MR 751133 | Zbl 0565.14002

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

[42] Namikawa, Y. Extension of 2-forms and symplectic varieties, J. Reine Angew. Math., Tome 539 (2001), pp. 123-147 | Article | MR 1863856 | Zbl 0996.53050

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

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

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

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

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

[48] O’Grady, K. Desingularized moduli spaces of sheaves on a K3, J. reine angew. Math., Tome 512 (1999), pp. 49-117 | Article | MR 1139878 | Zbl 0749.14030

[49] O’Grady, K. A new six-dimensional irreducible symplectic variety, J. Algebraic Geom., Tome 12 (2003), pp. 435-505 | Article | MR 1796694 | Zbl 1018.32028

[50] Panyushev, D. Rationality of singularities and the Gorenstein property for nilpotent orbits, Funct. Anal. Appl., Tome 25 (1991), p. 225-226 | Article | MR 1966025 | Zbl 02064089

[51] Verbitsky, M. Holomorphic symplectic geometry and orbifold singularities, Asian J. Math., Tome 4 (2000), pp. 553-563 | MR 2010734 | Zbl 1036.14007

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

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