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Multiple $$\zeta$$-motives and moduli spaces $$\overline{\mathcal M}_{0,n}$$. (English) Zbl 1047.11063
The authors give a beautiful geometric construction of framed Tate motives unramified over $$\mathbb{Z}$$ whose periods are the multiple zeta values. More precisely, they consider the moduli space $$\overline{{\mathcal M}}_{0,n+3}$$ of stable curves of genus zero with $$n+ 3$$ labeled points and define two boundary divisors $$A$$ and $$B$$ which share no common irreducible components. Then a mixed Tate motive over $$\mathbb{Z}$$ is provided by the $$n$$th cohomology $$H^n(\overline{{\mathcal M}}_{0,n+3}- A,B-A\cap B)$$ with a framing $$(\Omega_A,\Delta_B)$$ and the period is shown to be the multiple zeta value, where $$B$$ is an algebraic counterpart of the Stasheff polytope $$\Delta_B$$.

##### MSC:
 11G55 Polylogarithms and relations with $$K$$-theory 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11R32 Galois theory 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 14H10 Families, moduli of curves (algebraic)
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