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Let $$(X,H)$$ be a polarized smooth projective surface over the complex numbers which is symplectic, that is either $$K3$$ or abelian. Fix a primitive Mukai vector $$v \in H^{\text even}(X, \mathbb Z)$$ with $$\langle v, v \rangle =2$$.
Denote by $$M_{2v}$$ the moduli space of semistable sheaves on $$X$$ with respect to the polarization $$H$$ with Mukai vector $$2v$$.
The main result of the article is Theorem 1.1 which states that the blow up $$\tilde M_{2v} \to M_{2v}$$ of $$M_{2v}$$ along the reduced singular locus provides a symplectic resolution.
This simplifies the description of the symplectic varieties constructed by K. G. O’Grady [J. Reine Angew. Math. 512, 47–117 (1999; Zbl 0928.14029); J. Algebr. Geom. 12, No. 3, 435–505 (2003; Zbl 1068.53058)] which do not appear in Beauville’s two series.
The authors consider the stratification by automorphism type $$M_{2v} \supset S^2M_v \supset \Delta_{M_v}$$ of the moduli space $$M_{2v}$$. The dense open set $$M_{2v} \setminus S^2M_v$$ parameterizes stable sheaves $$E$$. The stratum $$S^2M_v \setminus \Delta_{M_v}$$ parameterizes polystable sheaves $$E=F_1 \oplus F_2$$ with $$[F_i] \in M_V$$ and $$F_1 \not \cong F_2$$. Finally the closed stratum $$\Delta_{M_v}$$ parameterizes sheaves $$F \oplus F$$ for $$[F] \in M_v$$.
On the other hand they consider the affine space $$Z$$ which is explicitly given as subvariety $$Z \subset \mathfrak {sp}(V,\omega)$$ where $$(V,\omega)$$ is a four dimensional symplectic vector space, and $$Z$$ consists of those $$B \in \mathfrak {sp}(V,\omega)$$ with $$B^2=0$$. $$Z$$ is singular but possesses a semi-small resolution $$\tilde Z \to Z$$ which can be described as a blow up (Theorem 2.1). The key result (Theorem 4.5) is that there exists an isomorphism of germs of analytic spaces $$(M_{2v},[F\oplus F] ) \cong (\mathbb C^4 \times Z ,0)$$. Thus, the resolution $$\tilde Z$$ of $$Z$$ yields a resolution $$\tilde M_{2v} \to M_{2v}$$.
Reviewer: Georg Hein (Essen)

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14J28 $$K3$$ surfaces and Enriques surfaces 14K10 Algebraic moduli of abelian varieties, classification 14D20 Algebraic moduli problems, moduli of vector bundles
SINGULAR
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##### References:
 [1] I. V. Artamkin, Deformation of torsion-free sheaves on an algebraic surface, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 435 – 468 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 449 – 485. · Zbl 0709.14013 [2] I. V. Artamkin, On the deformation of sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 660 – 665, 672 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 663 – 668. · Zbl 0661.14007 [3] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277 – 291. · Zbl 0172.05301 · doi:10.1007/BF01389777 · doi.org [4] Arnaud Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755 – 782 (1984) (French). · Zbl 0537.53056 [5] David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. · Zbl 0972.17008 [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] Phillip A. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 185 – 251. [8] Baohua Fu, Symplectic resolutions for nilpotent orbits, Invent. Math. 151 (2003), no. 1, 167 – 186. · Zbl 1072.14058 · doi:10.1007/s00222-002-0260-9 · doi.org [9] Mark Haiman, \?,\?-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), no. 1-3, 201 – 224. Selected papers in honor of Adriano Garsia (Taormina, 1994). · Zbl 1061.05509 · doi:10.1016/S0012-365X(98)00141-1 · doi.org [10] Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941 – 1006. · Zbl 1009.14001 [11] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. · Zbl 0872.14002 [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] Kieran G. O’Grady, Desingularized moduli spaces of sheaves on a \?3, J. Reine Angew. Math. 512 (1999), 49 – 117. · Zbl 0928.14029 · doi:10.1515/crll.1999.056 · doi.org [15] Kieran G. O’Grady, A new six-dimensional irreducible symplectic variety, J. Algebraic Geom. 12 (2003), no. 3, 435 – 505. · Zbl 1068.53058 [16] A. Rapagnetta, Topological invariants of O’Grady’s six dimensional irreducible symplectic variety. math.AG/0406026. · Zbl 1121.14014 [17] H. Weyl, The Classical Groups, Princeton Math. Series 1, Princeton 1946. [18] Kōta Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), no. 4, 817 – 884. · Zbl 1066.14013 · doi:10.1007/s002080100255 · doi.org [19] K. Yoshioka, A note on Fourier-Mukai transform, math.AG/0112267.
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