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Resolving mixed Hodge modules on configuration spaces. (English) Zbl 0986.14005

Let \(\pi:X\to S\) be a continuous map of locally compact topological spaces and \(n\) a natural number. Let \(X^n/S\) be the \(n\)-th fibred power of \(X\). Let \(F(X/S,n)\) be the configuration space whose fibre \(F(X/S,n)\) over a point \(s\in S\) is a configuration of \(n\) distinct points in the fibre \(X\), and let \(j(n): F(X/S,n)\to X^n/S\) be the natural open embedding. Given a sheaf \({\mathcal F}\) of abelian groups on \(X^n/S\), the author introduces a natural resolution of the sheaf \(j(n)_!j(n)^*{\mathcal F}\) by sums of terms of the form \(i(J)_!i(J)^*{\mathcal F}\), where \(i(J)\) is the closed embedding of a diagonal associated with a partition \(J\) of \([1,n)]\). This resolution has the property that if \({\mathcal F}\) is \(\mathfrak S_n\)-equivariant, so is the resolution, where the \(n\)-th symmetric group \(\mathfrak S_n\) acts on \(X^n/S\) naturally. The author then shows that a similar resolution exists when \(\pi:X\to S\) is a quasi-projective morphism of complex varieties and \({\mathcal F}\) is a mixed Hodge module. Applying this result, he obtains a formula for the relative \({\mathfrak S}_n\)-equivariant Serre characteristic of \({\mathcal M}_{1,n}/ {\mathcal M}_{1,1}\), where \({\mathcal M}_{1,n}\) is the quotient of \({\mathcal M}_{1,n}(N)\), the fine moduli space of elliptic curves of level \(N\geq 3\) with \(n\) marked points by the finite group \(\text{SL} (2,\mathbb{Z}/N)\).

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
14H10 Families, moduli of curves (algebraic)
11F12 Automorphic forms, one variable

Software:

SF
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Full Text: DOI arXiv

References:

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