Multiple zeta values and periods of moduli spaces ${\overline{𝔐}}_{0,n}$

Brown, Francis C. S.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 3, pp. 371-489.

Nous démontrons une conjecture de Goncharov et Manin qui prédit que les périodes des espaces de modules ${𝔐}_{0,n}$ des courbes de genre $0$ avec $n$ points marqués sont des valeurs zêta multiples. Nous introduisons une algèbre différentielle de fonctions polylogarithmes multiples sur ${𝔐}_{0,n}$ dans laquelle il existe des primitives. L’idée principale est d’appliquer une version de la formule de Stokes récursivement pour réduire chaque intégrale de périodes à une combinaison linéaire de valeurs zêta multiples. Nous donnons également une interprétation géométrique des double relations de mélange pour les valeurs zêta multiples. En considérant des applications naturelles entre les espaces des modules, on déduit des formules de produit générales entre leurs périodes. Les doubles relations de mélange s’obtiennent comme deux cas particuliers de cette construction.

We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces ${𝔐}_{0,n}$ of Riemann spheres with $n$ marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on ${𝔐}_{0,n}$ and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces.

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