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Baohua Fu
A survey on symplectic singularities and symplectic resolutions
Annales mathématiques Blaise Pascal, 13 no. 2 (2006), p. 209-236, doi: 10.5802/ambp.218
Article PDF | Reviews MR 2275448 | Zbl 1116.14008
[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, p. 1-32 MR 1672108
[2] V. Batyrev. Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc., 1:5-33, 1999.
Article | MR 1677693 | Zbl 0943.14004
[3] A. Beauville. Fano contact manifolds and nilpotent orbits. Comment. Math. Helv, 73:566-583, 1998.
Article | MR 1639888 | Zbl 0946.53046
[4] A. Beauville. Symplectic singularities. Invent. Math., 139:541-549, 2000.
Article | MR 1738060 | Zbl 0958.14001
[5] R. Bezrukavnikov and D. Kaledin. McKay equivalence for symplectic resolutions of singularities. Proc. Steklov Inst. Math., 246:13-33, 2004. MR 2101282 | Zbl 1137.14301
[6] A. Bialynicki-Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2), 98:480-497, 1973.
Article | MR 366940 | Zbl 0275.14007
[7] F. Bottacin. Poisson structures on moduli spaces of sheaves over Poisson surfaces. Invent. Math., 121:421-436, 1995.
Article | MR 1346215 | Zbl 0829.14019
[8] D. Burns, Y. Hu and T. Luo, HyperKähler manifolds and birational transformations in dimension 4, Vector bundles and representation theory (Columbia, MO, 2002), Amer. Math. Soc., 2003, p. 141-149 MR 1987745 | Zbl 01989688
[9] Y. Cho, Y. Miyaoka and N. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds, in Miyaoka, Mori, ed., Higher dimensional birational geometry (Kyoto, 1997), Math. Soc. Japan, 2002, p. 1-88 MR 1929791 | Zbl 1063.14065
[10] J. Choy and Y.-H. Kiem. Nonexistence of crepant resolution of some moduli spaces of sheaves on a K3 surface. math.AG/0407100, 2004
arXiv
[11] J. Choy and Y.-H. Kiem. On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface. Math. Z., 252:557-575, 2006.
Article | MR 2207759 | Zbl pre05013686
[12] A. M. Cohen. Finite quaternionic reflection groups. J. Algebra, 64:293-324, 1980.
Article | MR 579063 | Zbl 0433.20035
[13] D. Collingwood and W. Mc Govern. Nilpotent orbits in semi-simple Lie algebras. Van Nostrand Reinhold Co., 1993 MR 1251060 | Zbl 0972.17008
[14] S. Druel. Singularités symplectiques. J. Algebraic Geom., 13:427-439, 2004. MR 2047675 | Zbl 1068.32018
[15] B. Fu. Symplectic resolutions for coverings of nilpotent orbits. C. R. Acad. Sci., 336:159-162, 2003. MR 1969571 | Zbl 1068.14055
[16] B. Fu. Symplectic resolutions for nilpotent orbits. Invent. Math., 151:167-186, 2003.
Article | MR 1943745 | Zbl 1072.14058
[17] B. Fu. Symplectic resolutions for nilpotent orbits (II). C. R. Acad. Sci., 337:277-281, 2003. MR 2009121 | Zbl 1073.14547
[18] B. Fu. Birational geometry in codimension 2 of symplectic resolutions. math.AG/0409224, 2004
arXiv
[19] B. Fu. Extremal contractions, stratified Mukai flops and Springer maps. math.AG/0605431, 2006
arXiv
[20] B. Fu. Mukai flops and deformations of symplectic resolutions. Math. Z., 253:87-96, 2006.
Article | MR 2206638 | Zbl 1098.14009
[21] B. Fu and Y. Namikawa. Uniqueness of crepant resolutions and symplectic singularities. Ann. Inst. Fourier, 54:1-19, 2004.
Cedram | MR 2069119 | Zbl 1063.14018
[22] V. Ginzburg and D. Kaledin. Poisson deformations of symplectic quotient singularities. Adv. Math., 186:1-57, 2004.
Article | MR 2065506 | Zbl 1062.53074
[23] I. Gordon. Baby Verma modules for rational Cherednik algebras. Bull. London Math. Soc., 35:321-336, 2003.
Article | MR 1960942 | Zbl 1042.16017
[24] Robert M. Guralnick and J. Saxl. Generation of finite almost simple groups by conjugates. J. Algebra, 268:519-571, 2003.
Article | MR 2009321 | Zbl 1037.20016
[25] R. Hartshorne. Algebraic Geometry. Springer-Verlag, 1977 MR 463157 | Zbl 0367.14001
[26] W. Hesselink. Polarizations in the classical groups. Math. Z., 160:217-234, 1978.
Article | MR 480765 | Zbl 0364.20048
[27] Y. Hu. Geometric Invariant Theory and Birational Geometry. math.AG/0502462, 2005
arXiv
[28] Y. Hu and S.-T. Yau. HyperKähler manifolds and birational transformations. Adv. Theor. Math. Phys., 6:557-574, 2002. MR 1957670 | Zbl 1044.81105
[29] D. Huybrechts. Compact hyper-Kähler manifolds: basic results. Invent. Math., 135:63-113, 1999.
Article | MR 1664696 | Zbl 0953.53031
[30] D. Kaledin. Symplectic singularities from the Poisson point of view. J. Reine Angew. Math.
[31] D. Kaledin. Symplectic resolutions: deformations and birational maps. math.AG/0012008, 2000
arXiv
[32] D. Kaledin. McKay correspondence for symplectic quotient singularities. Invent. math., 148:150-175, 2002.
Article | MR 1892847 | Zbl 1060.14020
[33] D. Kaledin. On crepant resolutions of symplectic quotient singularities. Selecta Math. (N.S.), 9:529-555, 2003.
Article | MR 2031751 | Zbl 1066.14003
[34] D. Kaledin. Derived equivalence by quantization. math.AG/0504584, 2005
arXiv
[35] D. Kaledin and M. Lehn. Local structure of hyperKaehler singularities in O’Grady’s examples. math.AG/0405575, 2004
arXiv
[36] D. Kaledin, M. Lehn and C. Sorger. Singular symplectic moduli spaces. Invent. Math., 164:591-614, 2006.
Article | MR 2221132 | Zbl 1096.14037
[37] Y. Kawamata. D-equivalence and K-equivalence. J. Differential Geom., 61:147-171, 2002.
Article | MR 1949787 | Zbl 1056.14021
[38] H. Kraft and C. Procesi. Closures of conjugacy classes of matrices are normal. Invent. Math., 53:227-247, 1979.
Article | MR 549399 | Zbl 0434.14026
[39] E. Markman. Brill-Noether duality for moduli spaces of sheaves of $K3$ surfaces. J. Algebr. Geom., 10:623-694, 2001. MR 1838974 | Zbl 1074.14525
[40] S. Mukai. Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface. Invent. Math., 77:101-116, 1984.
Article | MR 751133 | Zbl 0565.14002
[41] Y. Namikawa. Deformation theory of singular symplectic n-folds. Math. Ann., 319:597-623, 2001.
Article | MR 1819886 | Zbl 0989.53055
[42] Y. Namikawa. Extension of 2-forms and symplectic varieties. J. Reine Angew. Math., 539:123-147, 2001.
Article | MR 1863856 | Zbl 0996.53050
[43] Y. Namikawa. A note on symplectic singularitie. math.AG/0101028, 2001
arXiv
[44] Y. Namikawa. Birational geometry of symplectic resolutions of nilpotent orbits. math.AG/0404072, 2004
arXiv
[45] Y. Namikawa. Birational geometry of symplectic resolutions of nilpotent orbits II. math.AG/0408274, 2004
arXiv
[46] Y. Namikawa. Flops and Poisson deformations of symplectic varieties. math.AG/0510059, 2005
arXiv
[47] Y. Namikawa. On deformations of Q-factorial symplectic varieties. math.AG/0506534, 2005
arXiv
[48] K. O’Grady. Desingularized moduli spaces of sheaves on a $K3$. J. reine angew. Math., 512:49-117, 1999.
Article | MR 1139878 | Zbl 0749.14030
[49] K. O’Grady. A new six-dimensional irreducible symplectic variety. J. Algebraic Geom., 12:435-505, 2003.
Article | MR 1796694 | Zbl 1018.32028
[50] D. Panyushev. Rationality of singularities and the Gorenstein property for nilpotent orbits. Funct. Anal. Appl., 25:225-226, 1991.
Article | MR 1966025 | Zbl 02064089
[51] M. Verbitsky. Holomorphic symplectic geometry and orbifold singularities. Asian J. Math., 4:553-563, 2000. MR 2010734 | Zbl 1036.14007
[52] J. Wierzba. Contractions of symplectic varieties. J. Algebraic Geom., 12:507-534, 2003.
Article
[53] J. Wierzba and J. A. Wisniewski. Small contractions of symplectic 4-folds. Duke Math. J., 120:65-95, 2003.
Article
Baohua Fu
A survey on symplectic singularities and symplectic resolutions
Annales mathématiques Blaise Pascal, 13 no. 2 (2006), p. 209-236, doi: 10.5802/ambp.218
Article PDF | Reviews MR 2275448 | Zbl 1116.14008
Résumé - Abstract
This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions.
Bibliography
[2] V. Batyrev. Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc., 1:5-33, 1999.
Article | MR 1677693 | Zbl 0943.14004
[3] A. Beauville. Fano contact manifolds and nilpotent orbits. Comment. Math. Helv, 73:566-583, 1998.
Article | MR 1639888 | Zbl 0946.53046
[4] A. Beauville. Symplectic singularities. Invent. Math., 139:541-549, 2000.
Article | MR 1738060 | Zbl 0958.14001
[5] R. Bezrukavnikov and D. Kaledin. McKay equivalence for symplectic resolutions of singularities. Proc. Steklov Inst. Math., 246:13-33, 2004. MR 2101282 | Zbl 1137.14301
[6] A. Bialynicki-Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2), 98:480-497, 1973.
Article | MR 366940 | Zbl 0275.14007
[7] F. Bottacin. Poisson structures on moduli spaces of sheaves over Poisson surfaces. Invent. Math., 121:421-436, 1995.
Article | MR 1346215 | Zbl 0829.14019
[8] D. Burns, Y. Hu and T. Luo, HyperKähler manifolds and birational transformations in dimension 4, Vector bundles and representation theory (Columbia, MO, 2002), Amer. Math. Soc., 2003, p. 141-149 MR 1987745 | Zbl 01989688
[9] Y. Cho, Y. Miyaoka and N. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds, in Miyaoka, Mori, ed., Higher dimensional birational geometry (Kyoto, 1997), Math. Soc. Japan, 2002, p. 1-88 MR 1929791 | Zbl 1063.14065
[10] J. Choy and Y.-H. Kiem. Nonexistence of crepant resolution of some moduli spaces of sheaves on a K3 surface. math.AG/0407100, 2004
arXiv
[11] J. Choy and Y.-H. Kiem. On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface. Math. Z., 252:557-575, 2006.
Article | MR 2207759 | Zbl pre05013686
[12] A. M. Cohen. Finite quaternionic reflection groups. J. Algebra, 64:293-324, 1980.
Article | MR 579063 | Zbl 0433.20035
[13] D. Collingwood and W. Mc Govern. Nilpotent orbits in semi-simple Lie algebras. Van Nostrand Reinhold Co., 1993 MR 1251060 | Zbl 0972.17008
[14] S. Druel. Singularités symplectiques. J. Algebraic Geom., 13:427-439, 2004. MR 2047675 | Zbl 1068.32018
[15] B. Fu. Symplectic resolutions for coverings of nilpotent orbits. C. R. Acad. Sci., 336:159-162, 2003. MR 1969571 | Zbl 1068.14055
[16] B. Fu. Symplectic resolutions for nilpotent orbits. Invent. Math., 151:167-186, 2003.
Article | MR 1943745 | Zbl 1072.14058
[17] B. Fu. Symplectic resolutions for nilpotent orbits (II). C. R. Acad. Sci., 337:277-281, 2003. MR 2009121 | Zbl 1073.14547
[18] B. Fu. Birational geometry in codimension 2 of symplectic resolutions. math.AG/0409224, 2004
arXiv
[19] B. Fu. Extremal contractions, stratified Mukai flops and Springer maps. math.AG/0605431, 2006
arXiv
[20] B. Fu. Mukai flops and deformations of symplectic resolutions. Math. Z., 253:87-96, 2006.
Article | MR 2206638 | Zbl 1098.14009
[21] B. Fu and Y. Namikawa. Uniqueness of crepant resolutions and symplectic singularities. Ann. Inst. Fourier, 54:1-19, 2004.
Cedram | MR 2069119 | Zbl 1063.14018
[22] V. Ginzburg and D. Kaledin. Poisson deformations of symplectic quotient singularities. Adv. Math., 186:1-57, 2004.
Article | MR 2065506 | Zbl 1062.53074
[23] I. Gordon. Baby Verma modules for rational Cherednik algebras. Bull. London Math. Soc., 35:321-336, 2003.
Article | MR 1960942 | Zbl 1042.16017
[24] Robert M. Guralnick and J. Saxl. Generation of finite almost simple groups by conjugates. J. Algebra, 268:519-571, 2003.
Article | MR 2009321 | Zbl 1037.20016
[25] R. Hartshorne. Algebraic Geometry. Springer-Verlag, 1977 MR 463157 | Zbl 0367.14001
[26] W. Hesselink. Polarizations in the classical groups. Math. Z., 160:217-234, 1978.
Article | MR 480765 | Zbl 0364.20048
[27] Y. Hu. Geometric Invariant Theory and Birational Geometry. math.AG/0502462, 2005
arXiv
[28] Y. Hu and S.-T. Yau. HyperKähler manifolds and birational transformations. Adv. Theor. Math. Phys., 6:557-574, 2002. MR 1957670 | Zbl 1044.81105
[29] D. Huybrechts. Compact hyper-Kähler manifolds: basic results. Invent. Math., 135:63-113, 1999.
Article | MR 1664696 | Zbl 0953.53031
[30] D. Kaledin. Symplectic singularities from the Poisson point of view. J. Reine Angew. Math.
[31] D. Kaledin. Symplectic resolutions: deformations and birational maps. math.AG/0012008, 2000
arXiv
[32] D. Kaledin. McKay correspondence for symplectic quotient singularities. Invent. math., 148:150-175, 2002.
Article | MR 1892847 | Zbl 1060.14020
[33] D. Kaledin. On crepant resolutions of symplectic quotient singularities. Selecta Math. (N.S.), 9:529-555, 2003.
Article | MR 2031751 | Zbl 1066.14003
[34] D. Kaledin. Derived equivalence by quantization. math.AG/0504584, 2005
arXiv
[35] D. Kaledin and M. Lehn. Local structure of hyperKaehler singularities in O’Grady’s examples. math.AG/0405575, 2004
arXiv
[36] D. Kaledin, M. Lehn and C. Sorger. Singular symplectic moduli spaces. Invent. Math., 164:591-614, 2006.
Article | MR 2221132 | Zbl 1096.14037
[37] Y. Kawamata. D-equivalence and K-equivalence. J. Differential Geom., 61:147-171, 2002.
Article | MR 1949787 | Zbl 1056.14021
[38] H. Kraft and C. Procesi. Closures of conjugacy classes of matrices are normal. Invent. Math., 53:227-247, 1979.
Article | MR 549399 | Zbl 0434.14026
[39] E. Markman. Brill-Noether duality for moduli spaces of sheaves of $K3$ surfaces. J. Algebr. Geom., 10:623-694, 2001. MR 1838974 | Zbl 1074.14525
[40] S. Mukai. Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface. Invent. Math., 77:101-116, 1984.
Article | MR 751133 | Zbl 0565.14002
[41] Y. Namikawa. Deformation theory of singular symplectic n-folds. Math. Ann., 319:597-623, 2001.
Article | MR 1819886 | Zbl 0989.53055
[42] Y. Namikawa. Extension of 2-forms and symplectic varieties. J. Reine Angew. Math., 539:123-147, 2001.
Article | MR 1863856 | Zbl 0996.53050
[43] Y. Namikawa. A note on symplectic singularitie. math.AG/0101028, 2001
arXiv
[44] Y. Namikawa. Birational geometry of symplectic resolutions of nilpotent orbits. math.AG/0404072, 2004
arXiv
[45] Y. Namikawa. Birational geometry of symplectic resolutions of nilpotent orbits II. math.AG/0408274, 2004
arXiv
[46] Y. Namikawa. Flops and Poisson deformations of symplectic varieties. math.AG/0510059, 2005
arXiv
[47] Y. Namikawa. On deformations of Q-factorial symplectic varieties. math.AG/0506534, 2005
arXiv
[48] K. O’Grady. Desingularized moduli spaces of sheaves on a $K3$. J. reine angew. Math., 512:49-117, 1999.
Article | MR 1139878 | Zbl 0749.14030
[49] K. O’Grady. A new six-dimensional irreducible symplectic variety. J. Algebraic Geom., 12:435-505, 2003.
Article | MR 1796694 | Zbl 1018.32028
[50] D. Panyushev. Rationality of singularities and the Gorenstein property for nilpotent orbits. Funct. Anal. Appl., 25:225-226, 1991.
Article | MR 1966025 | Zbl 02064089
[51] M. Verbitsky. Holomorphic symplectic geometry and orbifold singularities. Asian J. Math., 4:553-563, 2000. MR 2010734 | Zbl 1036.14007
[52] J. Wierzba. Contractions of symplectic varieties. J. Algebraic Geom., 12:507-534, 2003.
Article
[53] J. Wierzba and J. A. Wisniewski. Small contractions of symplectic 4-folds. Duke Math. J., 120:65-95, 2003.
Article
© 2013
Annales Mathématiques Blaise Pascal
Published by the
Laboratoire de mathématiques
CNRS - UMR 6620
Université Blaise Pascal de Clermont-Ferrand
Annales Mathématiques Blaise Pascal
Published by the
Laboratoire de mathématiques
CNRS - UMR 6620
Université Blaise Pascal de Clermont-Ferrand
