My research interests concern mainly the theory of Irreducible Symplectic Varieties, both smooth and singular, with particular attention to those that arise from moduli spaces of sheaves on K3 and Abelian surfaces. At the moment, I'm focused on problems regarding the bimeromorphic classification of such varieties, related to and motivated by Torelli Theorem for irreducible symplectic varieties.
The subject of my PhD Thesis was the description of the locally trivial monodromy group of singular moduli spaces of sheaves on Abelian surfaces.
Further topics I'm currently interested in include:
Birationality of moduli spaces of sheaves on Abelian surfaces and relation with wall-crossing for Bridgeland Stability conditions.
Moduli spaces of sheaves on hyperkähler manifolds.
Fano fourfolds of K3 type and link with hyperkähler geometry.
Preprints
Locally trivial monodromy of moduli spaces of sheaves on Abelian surfaces.
Ludovica Buelli. ArXiv preprint 2025 (arXiv:2510.23193v2).
Abstract: The aim of this work is to give a description of the locally trivial monodromy group of irreducible symplectic varieties arising from moduli spaces of semistable sheaves on Abelian surfaces with non-primitive Mukai vector. The outcome is that the locally trivial monodromy group of a singular moduli space of this type is isomorphic to the monodromy group of a smooth moduli space, extending Markman’s and Mongardi's description to the non-primitive case. As a consequence, we also prove the SYZ conjecture for any singular moduli space of this type.
On moduli spaces of vector bundles on $K3^{[n]}-$type IHS manifolds.
Nicolò Bignami, Ludovica Buelli, Irene Macías Tarrío, Roberto Wiktor Vacca, Vanja Zuliani. ArXiv preprint 2026 (arXiv:2606.23622).
Abstract: We study moduli spaces of modular vector bundles on projective irreducible holomorphic symplectic manifolds of $K3^{[n]}-$type. Under suitable numerical assumptions, we exhibit connected components of these moduli spaces which are again irreducible holomorphic symplectic manifolds of $K3^{[n]}-$type. Moreover, the corresponding universal families induce derived equivalences with the original manifolds. This produces smooth components of moduli spaces of modular vector bundles on irreducible holomorphic symplectic manifolds of any even dimension.
PhD Thesis
Locally trivial monodromy of moduli spaces of sheaves on Abelian surfaces.
Ludovica Buelli, University of Genoa, 2026, soon available at https://hdl.handle.net/11567/1301457.
Abstract: 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.