Stable cohomology of the universal Picard varieties and the extended mapping class group

TL;DR

This paper investigates the cohomology of moduli spaces of surfaces with line bundles, relating them to extended mapping class groups and complex algebraic geometry, providing new insights into their structure and invariants.

Contribution

It establishes connections between the cohomology of universal Picard varieties, extended mapping class groups, and complex algebraic geometry, with explicit calculations and constructions.

Findings

Homological stability results for moduli spaces with line bundles

Cohomological calculations derived from stability and Madsen--Weiss type results

Construction of a holomorphic stack and computation of its Picard group

Abstract

We study the moduli spaces which classify smooth surfaces along with a complex line bundle. There are homological stability and Madsen--Weiss type results for these spaces (mostly due to Cohen and Madsen), and we discuss the cohomological calculations which may be deduced from them. We then relate these spaces to (a generalisation of) Kawazumi's extended mapping class groups, and hence deduce cohomological information about these. Finally, we relate these results to complex algebraic geometry. We construct a holomorphic stack classifying families of Riemann surfaces equipped with a fibrewise holomorphic line bundle, which is a gerbe over the universal Picard variety, and compute its holomorphic Picard group.