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The Beauville-Bogomolov lattice is computed for a simplest singularsymplectic manifold of dimension 4, obtained as a partial desingularization ofthe quotient $S^{[2]}/\iota$, where $S^{[2]}$ is the Hilbert square of a K3surface $S$ and $\iota$ is a symplectic involution on it. This result applies,in particular, to the singular symplectic manifolds of dimension 4, constructedby Markushevich-Tikhomirov as compactifications of families of Prym varietiesof a linear system of curves on a K3 surface with an anti-symplecticinvolution.

Beauville-Bogomolov lattice for a singular symplectic variety of dimension 4