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

Johannes Ebert; Oscar Randal-Williams

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

Institut for Matematiske Fag

7 Citationer (Scopus)

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.

Originalsprog	Engelsk
Tidsskrift	Documenta Mathematica
Vol/bind	17
Sider (fra-til)	417-450
ISSN	1431-0635
Status	Udgivet - 2012

Citationsformater

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T1 - Stable cohomology of the universal Picard varieties and the extended mapping class group

AU - Ebert, Johannes

AU - Randal-Williams, Oscar

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N2 - 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.

AB - 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.

M3 - Journal article

SN - 1431-0635

VL - 17

SP - 417

EP - 450

JO - Documenta Mathematica

JF - Documenta Mathematica

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