A survey on symplectic singularities and symplectic resolutions

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

Annales mathématiques Blaise Pascal (2006)

  • Volume: 13, Issue: 2, page 209-236
  • ISSN: 1259-1734


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

How to cite


Fu, Baohua. "A survey on symplectic singularities and symplectic resolutions." Annales mathématiques Blaise Pascal 13.2 (2006): 209-236. <http://eudml.org/doc/10531>.

abstract = {This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions.},
affiliation = {Laboratoire J. Leray Université de Nantes, Faculté des sciences 2, Rue de la Houssinière BP 92208, F-44322 Nantes Cedex 03 France},
author = {Fu, Baohua},
journal = {Annales mathématiques Blaise Pascal},
keywords = {symplectic variety; symplectic resolution; birational geometry},
language = {eng},
month = {7},
number = {2},
pages = {209-236},
publisher = {Annales mathématiques Blaise Pascal},
title = {A survey on symplectic singularities and symplectic resolutions},
url = {http://eudml.org/doc/10531},
volume = {13},
year = {2006},

AU - Fu, Baohua
TI - A survey on symplectic singularities and symplectic resolutions
JO - Annales mathématiques Blaise Pascal
DA - 2006/7//
PB - Annales mathématiques Blaise Pascal
VL - 13
IS - 2
SP - 209
EP - 236
AB - This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions.
LA - eng
KW - symplectic variety; symplectic resolution; birational geometry
UR - http://eudml.org/doc/10531
ER -


  1. V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) (1998), 1-32, Publish or Perish, Inc., Houston Zbl0963.14015MR1672108
  2. V. Batyrev, Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. 1 (1999), 5-33 Zbl0943.14004MR1677693
  3. A. Beauville, Fano contact manifolds and nilpotent orbits, Comment. Math. Helv 73 (1998), 566-583 Zbl0946.53046MR1639888
  4. A. Beauville, Symplectic singularities, Invent. Math. 139 (2000), 541-549 Zbl0958.14001MR1738060
  5. R. Bezrukavnikov, D. Kaledin, McKay equivalence for symplectic resolutions of singularities, Proc. Steklov Inst. Math. 246 (2004), 13-33 Zbl1137.14301MR2101282
  6. A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480-497 Zbl0275.14007MR366940
  7. F. Bottacin, Poisson structures on moduli spaces of sheaves over Poisson surfaces, Invent. Math. 121 (1995), 421-436 Zbl0829.14019MR1346215
  8. D. Burns, Y. Hu, T. Luo, HyperKähler manifolds and birational transformations in dimension 4, Vector bundles and representation theory (Columbia, MO, 2002) (2003), 141-149, Amer. Math. Soc. Zbl1080.14508MR1987745
  9. Y. Cho, Y. Miyaoka, N. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997) (2002), 1-88, Miyaoka Zbl1063.14065MR1929791
  10. J. Choy, Y.-H. Kiem, Nonexistence of crepant resolution of some moduli spaces of sheaves on a K3 surface, (2004) Zbl1126.14051
  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. 252 (2006), 557-575 Zbl05013686MR2207759
  12. A. M. Cohen, Finite quaternionic reflection groups, J. Algebra 64 (1980), 293-324 Zbl0433.20035MR579063
  13. D. Collingwood, W. Mc Govern, Nilpotent orbits in semi-simple Lie algebras, (1993), Van Nostrand Reinhold Co., New York Zbl0972.17008MR1251060
  14. S. Druel, Singularités symplectiques, J. Algebraic Geom. 13 (2004), 427-439 Zbl1068.32018MR2047675
  15. B. Fu, Symplectic resolutions for coverings of nilpotent orbits, C. R. Acad. Sci. 336 (2003), 159-162 Zbl1068.14055MR1969571
  16. B. Fu, Symplectic resolutions for nilpotent orbits, Invent. Math. 151 (2003), 167-186 Zbl1072.14058MR1943745
  17. B. Fu, Symplectic resolutions for nilpotent orbits (II), C. R. Acad. Sci. 337 (2003), 277-281 Zbl1073.14547MR2009121
  18. B. Fu, Birational geometry in codimension 2 of symplectic resolutions, (2004) 
  19. B. Fu, Extremal contractions, stratified Mukai flops and Springer maps, (2006) Zbl1120.14039
  20. B. Fu, Mukai flops and deformations of symplectic resolutions, Math. Z. 253 (2006), 87-96 Zbl1098.14009MR2206638
  21. B. Fu, Y. Namikawa, Uniqueness of crepant resolutions and symplectic singularities, Ann. Inst. Fourier 54 (2004), 1-19 Zbl1063.14018MR2069119
  22. V. Ginzburg, D. Kaledin, Poisson deformations of symplectic quotient singularities, Adv. Math. 186 (2004), 1-57 Zbl1062.53074MR2065506
  23. I. Gordon, Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc. 35 (2003), 321-336 Zbl1042.16017MR1960942
  24. Robert M. Guralnick, J. Saxl, Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), 519-571 Zbl1037.20016MR2009321
  25. R. Hartshorne, Algebraic Geometry, (1977), Springer-Verlag Zbl0367.14001MR463157
  26. W. Hesselink, Polarizations in the classical groups, Math. Z. 160 (1978), 217-234 Zbl0364.20048MR480765
  27. Y. Hu, Geometric Invariant Theory and Birational Geometry, (2005) 
  28. Y. Hu, S.-T. Yau, HyperKähler manifolds and birational transformations, Adv. Theor. Math. Phys. 6 (2002), 557-574 Zbl1044.81105MR1957670
  29. D. Huybrechts, Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), 63-113 Zbl0953.53031MR1664696
  30. D. Kaledin, Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. Zbl1121.53056
  31. D. Kaledin, Symplectic resolutions: deformations and birational maps, (2000) 
  32. D. Kaledin, McKay correspondence for symplectic quotient singularities, Invent. math. 148 (2002), 150-175 Zbl1060.14020MR1892847
  33. D. Kaledin, On crepant resolutions of symplectic quotient singularities, Selecta Math. (N.S.) 9 (2003), 529-555 Zbl1066.14003MR2031751
  34. D. Kaledin, Derived equivalence by quantization, (2005) Zbl1149.14009
  35. D. Kaledin, M. Lehn, Local structure of hyperKaehler singularities in O’Grady’s examples, (2004) Zbl1160.14006
  36. D. Kaledin, M. Lehn, C. Sorger, Singular symplectic moduli spaces, Invent. Math. 164 (2006), 591-614 Zbl1096.14037MR2221132
  37. Y. Kawamata, D-equivalence and K-equivalence, J. Differential Geom. 61 (2002), 147-171 Zbl1056.14021MR1949787
  38. H. Kraft, C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 (1979), 227-247 Zbl0434.14026MR549399
  39. E. Markman, Brill-Noether duality for moduli spaces of sheaves of K 3 surfaces, J. Algebr. Geom. 10 (2001), 623-694 Zbl1074.14525MR1838974
  40. S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface, Invent. Math. 77 (1984), 101-116 Zbl0565.14002MR751133
  41. Y. Namikawa, Deformation theory of singular symplectic n-folds, Math. Ann. 319 (2001), 597-623 Zbl0989.53055MR1819886
  42. Y. Namikawa, Extension of 2-forms and symplectic varieties, J. Reine Angew. Math. 539 (2001), 123-147 Zbl0996.53050MR1863856
  43. Y. Namikawa, A note on symplectic singularitie, (2001) 
  44. Y. Namikawa, Birational geometry of symplectic resolutions of nilpotent orbits, (2004) Zbl1117.14018
  45. Y. Namikawa, Birational geometry of symplectic resolutions of nilpotent orbits II, (2004) Zbl1117.14018
  46. Y. Namikawa, Flops and Poisson deformations of symplectic varieties, (2005) Zbl1148.14008
  47. Y. Namikawa, On deformations of Q-factorial symplectic varieties, (2005) Zbl1122.14029
  48. K. O’Grady, Desingularized moduli spaces of sheaves on a K 3 , J. reine angew. Math. 512 (1999), 49-117 Zbl0749.14030MR1139878
  49. K. O’Grady, A new six-dimensional irreducible symplectic variety, J. Algebraic Geom. 12 (2003), 435-505 Zbl1018.32028MR1796694
  50. D. Panyushev, Rationality of singularities and the Gorenstein property for nilpotent orbits, Funct. Anal. Appl. 25 (1991), 225-226 Zbl1094.14010MR1966025
  51. M. Verbitsky, Holomorphic symplectic geometry and orbifold singularities, Asian J. Math. 4 (2000), 553-563 Zbl1036.14007MR2010734
  52. J. Wierzba, Contractions of symplectic varieties, J. Algebraic Geom. 12 (2003), 507-534 Zbl1094.14010
  53. J. Wierzba, J. A. Wisniewski, Small contractions of symplectic 4-folds, Duke Math. J. 120 (2003), 65-95 Zbl1036.14007

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.