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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 $$\ell$$.

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