The SYZ conjecture for singular moduli spaces of sheaves on K3 surfaces
Claudio Onorati, Ángel David Ríos Ortiz
TL;DR
This work extends the SYZ conjecture to singular irreducible symplectic varieties that are locally trivially deformation equivalent to moduli spaces of sheaves on K3 surfaces. By generalizing lagrangian-fibration results to the singular setting, the authors show that nef isotropic classes arise from lagrangian fibrations, and establish openness and density of the lagrangian locus in the moduli of such varieties. They deduce that the Huybrechts–Riemann–Roch polynomial for these deformation types is of K3$^{[n]}$-type with $n=km^2+1$, and they compute the relative theta divisor on Beauville–Mukai-type moduli spaces, including its square and divisibility. A further corollary is that the polarisation type of lagrangian fibrations on these spaces is the primitive, principal type $(1,\,oldsymbol{1})$, confirming a unified picture across the smooth and singular cases and highlighting the role of Beauville–Mukai systems as deformation anchors.
Abstract
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.