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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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