Projective orbifolds of Nikulin type

We study projective irreducible symplectic orbifolds of dimension four that are deformations of partial resolutions of quotients of hyperkähler manifolds of $K{3}^{[2]}$-type by symplectic involutions; we call them orbifolds of Nikulin type. We first classify those projective orbifolds that are really quotients, by describing all families of projective fourfolds of $K{3}^{[2]}$-type with a symplectic involution and the relation with their quotients, and then study their deformations. We compute the Riemann–Roch formula for Weil divisors on orbifolds of Nikulin type and using this we describe the first known locally complete family of singular irreducible symplectic varieties as double covers of special complete intersections $(3,4)$ in ${\mathbb{P}}^6$.