The SYZ conjecture for singular moduli spaces of sheaves on K3 surfaces

In this paper we prove the SYZ conjecture for irreducible symplectic varieties that are locally trivial deformation equivalent to moduli spaces of sheaves on K3 surfaces. As an intermediate step in the argument, we generalise to the singular setting a result of Kamenova–Verbitsky and Matsushita about moduli spaces of lagrangian fibrations of primitive symplectic varieties. Two further corollaries are also presented: the computation of the Huybrechts–Riemann–Roch polynomial and of the polarisation type of this kind of symplectic varieties.