arXiv 1012.09012010-12math.ATmath.AG

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

Johannes Ebert, Oscar Randal-Williams

发表信息：: Doc. Math. 17 (2012), 417--450

英文摘要：

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.

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