Locally trivial monodromy of moduli spaces of sheaves on Abelian surfaces

The locally trivial monodromy group is an important locally trivial deformation invariant for irreducible symplectic varieties and plays a fundamental role in their bimeromorphic classification, by Global Torelli Theorem. While this group has been determined for all known deformation classes in the smooth case, this problem has only been partially addressed in the singular setting. This Thesis contributes to completing this picture by explicitly computing the locally trivial monodromy group for a distinguished and rich class of irreducible symplectic varieties, namely singular moduli spaces of sheaves on Abelian surfaces. We establish a lattice-theoretic description of this group and provide a clear geometric interpretation of the latter: we prove that it is isomorphic to the classical monodromy group of a smooth moduli space of sheaves of the same kind, embedded within the most singular locus of the singular moduli space. Moreover, its generators are explicitly described as isometries induced by monodromy operators of the underlying surface and certain Fourier-Mukai equivalences on the derived category of the latter. Finally, as a main geometric application of the monodromy description, we prove the SYZ conjecture for this locally trivial deformation class of singular symplectic varieties, showing that any nef and isotropic line bundle induces a Lagrangian fibration.