### Geometry of algebraic structures: moduli, invariants, deformations

Project Type: Prin 2022

Funded by: MUR

Period: Sep 28, 2023 – Sep 27, 2025

Budget: €77.847,00

Principal Investigator: Ugo Bruzzo (SISSA, Trieste)

Local coordinator: Rita Pardini (Università di Pisa)

##### Participants

Mattia Talpo (Università di Pisa), Tamás Szamuely (Università di Pisa), Marco Franciosi (Università di Pisa), Enrico Sbarra (Università di Pisa), Gregory James Pearlstein (Università di Pisa)

##### Description

The main topic of the project will be explored along several different lines. Moduli spaces, geometric and enumerative invariants, and deformations are the leading threads of the project, connecting the various directions in which it will develop. The different lines of investigations share several fundamental techniques and goals and they constitute a natural common ground for collaborations among the members of the project.
The main lines will be the following:
1. Mirror symmetry and quantum cohomology: homological mirror symmetry for complete intersections in algebraic tori, logarithmic Calabi-Yau varieties and scattering diagrams, smoothing of toric Fano varieties, crepant resolution conjecture.
2. Zero-dimensional Hilbert and Quot schemes and their motives, derived categories: motives of Quot schemes, generalized Hilbert-Chow morphisms, super-Nori motives.
3. Deformations, Moduli, Hodge Theory: Moduli of Q-Gorenstein surfaces, Hodge theory of boundaries of moduli spaces of surfaces, degeneration of Calabi-Yau threefolds fibered by elliptic K3 surfaces, Severi varieties of nodal surfaces, moduli of quiver representations and applications.