Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

La singularité de O'Grady


Authors: Manfred Lehn and Christoph Sorger
Journal: J. Algebraic Geom. 15 (2006), 753-770
DOI: https://doi.org/10.1090/S1056-3911-06-00437-1
Published electronically: May 24, 2006
MathSciNet review: 2237269
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Abstract | References | Additional Information

Abstract: Let $ M_{2v}$ be the moduli space of semistable sheaves with Mukai vector $ 2v$ on an abelian or $ K3$ surface where $ v$ is primitive such that $ \langle v,v \rangle=2$. We show that the blow-up of the reduced singular locus of $ M_{2v}$ provides a symplectic resolution of singularities. This provides a direct description of O'Grady's resolutions of $ M_{K3}(2,0,4)$ and $ M_{Ab}(2,0,2)$.

Résumé. Soit $ M_{2v}$ l'espace de modules des faisceaux semi-stables de vecteur de Mukai $ 2v$ sur une surface $ K3$ ou abélienne où $ v$ est primitif tel que $ \langle v,v \rangle=2$. Nous montrons que l'éclatement de $ M_{2v}$ le long de son lieu singulier réduit fournit une résolution symplectique des singularités. Ceci donne une description directe des résolutions de O'Grady de $ M_{K3}(2,0,4)$ et $ M_{Ab}(2,0,2)$.


References [Enhancements On Off] (What's this?)

  • 1. I. V. Artamkin, Deformation of torsion-free sheaves on an algebraic surface. (En russe) Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 435-468; (traduction anglaise) Math. USSR-Izv. 36 (1991), 449-485. MR 1072690 (91j:14010)
  • 2. I. V. Artamkin, On the deformation of sheaves. (En russe) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 660-665, 672; (traduction anglaise) Math. USSR-Izv. 32 (1989), 663-668. MR 0954302 (89g:14004)
  • 3. M. Artin, On the solutions of Analytic Equations. Invent. math. 5 (1968), 277-291. MR 0232018 (38:344)
  • 4. A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differential Geom. 18 (1983), 755-782. MR 0730926 (86c:32030)
  • 5. D. H. Collingwood, W. M. McGovern, Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Co., New York, 1993. xiv+186 pp. MR 1251060 (94j:17001)
  • 6. G.-M. Greuel, G. Pfister, and H. Schönemann. SINGULAR 2.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de
  • 7. P. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles. in: Global Analysis, papers in honor of K. Kodaira, pp. 185-251, Princeton University Press 1969. MR 0258070 (41:2717)
  • 8. B. Fu, Symplectic resolutions for nilpotent orbits. Invent. Math. 151 (2003), 167-186. MR 1943745 (2003j:14061)
  • 9. M. Haiman, t,q-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), no. 1-3, 201-224, Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661369 (2000k:05264)
  • 10. M. Haiman, Hilbert schemes, polygraphs, and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941-1006. MR 1839919 (2002c:14008)
  • 11. D. Huybrechts, M. Lehn, The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics E31, Vieweg Verlag, 1997. MR 1450870 (98g:14012)
  • 12. D. Kaledin, M. Lehn, Ch. Sorger, Singular Symplectic Moduli Spaces. À paraître dans Invent. Math.
  • 13. S. Kleiman, Les théorèmes de finititude pour le foncteur de Picard, SGA 6, Exp. 13, Springer Lecture Notes 225, 1971.
  • 14. K. O'Grady, Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math. 512 (1999), 49-117. MR 1703077 (2000f:14066)
  • 15. K. O'Grady, A new six-dimensional irreducible symplectic variety. J. Algebraic Geom. 12 (2003), 435-505. MR 1966024 (2004c:14017)
  • 16. A. Rapagnetta, Topological invariants of O'Grady's six dimensional irreducible symplectic variety. math.AG/0406026.
  • 17. H. Weyl, The Classical Groups, Princeton Math. Series 1, Princeton 1946.
  • 18. K. Yoshioka, Moduli spaces of stable sheaves on abelian surfaces. Math. Annalen 321 (2001), 817-884. MR 1872531 (2002k:14020)
  • 19. K. Yoshioka, A note on Fourier-Mukai transform, math.AG/0112267.


Additional Information

Manfred Lehn
Affiliation: Institut für Mathematik, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany
Email: lehn@mathematik.uni-mainz.de

Christoph Sorger
Affiliation: Institut Universitaire de France & Laboratoire de Mathématiques Jean Leray (UMR 6629 du CNRS), Université de Nantes, 2, Rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 03, France
Email: christoph.sorger@univ-nantes.fr

DOI: https://doi.org/10.1090/S1056-3911-06-00437-1
Received by editor(s): April 22, 2005
Received by editor(s) in revised form: September 29, 2005, and November 3, 2005
Published electronically: May 24, 2006

American Mathematical Society