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In this paper we classify all singular irreducible symplectic surfaces, i. e. , compact, connected complex surfaces with canonical singularities that have a holomorphic symplectic form on the smooth locus, and for which every finite quasi-\'etale covering has the algebra of reflexive forms spanned by the reflexive pull-back of. We moreover prove that the Hilbert scheme of two points on such a surface X is an irreducible symplectic variety, at least in the case where the smooth locus of X is simply connected.
Garbagnati et al. (Tue,) studied this question.
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