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Multiple zeta values and periods of moduli spaces \(\bar {\mathfrak M}_{0,n}\). (Périodes des espaces des modules \(\bar {\mathfrak M}_{0,n}\) et valeurs zêtas multiples.) (French) Zbl 1105.11019
Let \({\mathfrak M}_{0,n}\) denote the moduli space of genus 0 curves with \(n\) marked points. Let \(\overline{\mathfrak M}_{0,n}\) denote the smooth compactification of \({\mathfrak M}_{0,n}\) constructed by Deligne, Knudsen and Mumford. Fix two boundary divisors \(A\) and \(B\) (i.e., \(A,B \subset \overline{{\mathfrak M}}_{0,n}\setminus {\mathfrak M}_{0,n}\)) that have no irreducible component in common. A. B. Goncharov and Yu. I. Manin [Compos. Math. 140, No. 1, 1–14 (2004; Zbl 1047.11063)] showed that \(H^\ell(\overline{{\mathfrak M}}_{0,n}\setminus A, B\setminus B\cup A)\) is a mixed Tate motive unramified over \(\mathbb Z\). Fix positive integers \(n_1,\dots, n_r\) such that \(n_r>1\). Then \[ \zeta(n_1,\dots,n_r):=\sum_{0< k_1<\dots<k_r} \frac{1}{k_1^{n_1}\dots k_r^{n_r}} \] is called a multiple zeta value. Its weight is the quantity \(n_1+\dots+n_r\). A very general conjecture claims that the periods associated with mixed Tate motives unramified over \(\mathbb Z\) are multiple zeta values.
The paper under review sketches a proof for this conjecture for the mixed Tate motives
\(H^\ell(\overline{{\mathfrak M}}_{0,n}\setminus A, B\setminus B\cup A)\). The proof is highly topological. It uses that \(M_{0,n}(\mathbb R)\) can be tessellated by a number of open cells \(X_n\) which can be identified with a Stasheff polytope. Using a version of Stokes’ Theorem and induction the author relates the periods with multiple polylogarithms evaluated at 1, yielding that the periods are \(\mathbb Q[2\pi i]\)-linear combinations of multiple zeta values of weight at most \(\ell\).

11M32 Multiple Dirichlet series and zeta functions and multizeta values
11G55 Polylogarithms and relations with \(K\)-theory
14H10 Families, moduli of curves (algebraic)
Full Text: DOI
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