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

The Beauville-Bogomolov lattice is computed for a simplest singular symplectic manifold of dimension 4, obtained as a partial desingularization of the quotient S[2]/ι, where S[2] is the Hilbert square of a K3 surface S and ι is a symplectic involution on it. This result applies, in particular, to the singular symplectic manifolds of dimension 4, constructed by Markushevich-Tikhomirov as compactifications of families of Prym varieties of a linear system of curves on a K3 surface with an anti-symplectic involution.