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Desingularized moduli spaces of sheaves on a $$K3$$. (English) Zbl 0928.14029
Let $${\mathcal M}_c$$ be the moduli space of semistable rank-two torsion-free sheaves on a projective K3 surface $$X$$, with $$c_1=0$$ and $$c_2= c$$. We assume $$c$$ is even, and $$c\geq 4$$. Then (if the polarization is “generic”) $${\mathcal M}_c$$ is an irreducible projective variety of dimension $$(4c-6)$$, singular exactly along the locus parametrizing strictly semistable sheaves, i.e. of the form $$I_W\oplus I_Z$$, where $$2\ell(W)= 2\ell (Z)=c$$.
In the first part of the paper we construct a desingularization $$\widetilde{\mathcal M}_c$$ of $${\mathcal M}_c$$ by analyzing explicitly Kirwan’s procedure for desingularizing G.I.T. quotients. Then we consider the case $$c=4$$: We blow down $${\mathcal M}_4$$ in order to obtain a (holomorphic) symplectic desingularization $$\widetilde{\mathcal M}_4$$ of $${\mathcal M}_4$$. We then show that $$\widetilde{\mathcal M}_4$$ belongs to a new deformation class of irreducible symplectic manifolds (we prove $$b_2(\widetilde {{\mathcal M}}_4)\geq 24)$$.

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14J28 $$K3$$ surfaces and Enriques surfaces 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14J10 Families, moduli, classification: algebraic theory
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