The geometry of strata of differentials.

• Applicant: Dr. Matteo Costantini
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 520961294

The moduli space of Abelian differentials parametrizes Riemann surfaces of a fixed genus together with a holomorphic one form. By fixing the orders of the zeroes of the differential, we obtain a stratification of this moduli space whose strata are important objects studied from different mathematical perspectives. These strata appear naturally in many areas of mathematics, such as the study of flat surfaces, Teichmueller dynamics and they are associated to many counting problems. Despite various known results, the global geometry of these strata of differentials remains quite mysterious. Recent discoveries of the applicant with coauthors shed some light on some geometrical invariants in the case of abelian holomorphic differentials, like the Euler characteristic and the Kodaira dimension. One of the main new ingredients that allowed these results is the existence of a nice compactification of these strata given by the space of multi-scale differentials. The aim of the proposed project is to further investigate the geometry of strata of differentials via the use of the above mentioned compactification.

• DFG Programme: Research Grants