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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 \(b_2(\widetilde {{\mathcal M}}_4)\geq 24)\).

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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