Geometry and topology of fine compactified universal Jacobians

The project will be to study the cohomology of compactified universal Jacobians as a vector space, in the same way as was done for the moduli spaces of stable pointed curves Mbargn.

A successful method to calculate the cohomology is via Deligne's "Yoga of weights". The idea is that the usual, integer-valued (compactly supported) Euler characteristic is additive under a decomposition of a topological space into an open subset and its complement. For algebraic varieties, this additivity remains valid for refined versions of the Euler characteristics that "incorporate the weights", for example for the Euler characteristic that takes values in the category of Mixed Hodge structures. If the variety under analysis is smooth and compact, then by purity of the Hodge structures, the knowledge of the latter Euler characteristic is equivalent to knowing the Betti numbers.

The moduli spaces of stable pointed curves admit a stratification whose strata consist of moduli spaces with lower genus and points. This stratification has been extensively used to calculate the cohomology of the moduli spaces of curves "inductively" starting from low genus, using the method outlined in the previous paragraph. We propose a similar approach for the universal compactified Jacobian: a smooth and compact space fibered over the moduli space of stable curves, whose fiber over each smooth curve is the Jacobian variety of that curve. We propose looking at: (a) Explicit calculations in low genus, starting in genus 2 (the genus 1 case is covered by one of the main results of the recent paper https://arxiv.org/abs/2012.09142 by Pagani-Tommasi). (b) Trying to detect general structure, as in the theory of "Modular Operads" by Getzler-Kapranov. (Fine compactified universal Jacobians can be interpreted as "admissible G-covers" for G the multiplicative group C\{0}, and the case when G is finite was addressed in work by Jarvis-Kauffman-Kimura https://arxiv.org/abs/math/0302316, and Petersen https://arxiv.org/pdf/1205.0420.pdf.)