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Annales scientifiques de l'ENS - Titles - série 4, 50 (2017)

Titles < série 4, 50

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 50, fascicule 5 (2017)

Johan Alm, Dan Petersen
Brown's dihedral moduli space and freedom of the gravity operad
Annales scientifiques de l'ENS 50, fascicule 5 (2017), 1081-1122

Télécharger cet article : Fichier PDF

Résumé :
Espace de modules dièdre de Brown et liberté de l'opérade de gravité
Francis Brown a introduit une compactification partielle M_0,n^ de l'espace de modules M_0,n. Nous démontrons que la coopérade gravité, définie par la cohomologie (décalée en degré) des espaces M_0,n, est colibre comme coopérade non symétrique anti-cyclique; de plus, les cogénérateurs sont donnés par les groupes de cohomologie de M_0,n^. La preuve construit une base explicite de H^(M_0,n) en termes de diagrammes. Cette base est compatible avec la cocomposition coopéradique, et admet un sous-ensemble qui est une base de H^(M_0,n^). Nous montrons que nos résultats sont équivalents au fait que H^k(M_0,n^) a une structure de Hodge pure de poids 2k pour tout k, et nous donnons de plus dans notre article une seconde preuve, plus directe, de ce dernier fait. Cette seconde preuve utilise une construction itérative nouvelle et explicite de M_0,n^ à partir de A^n-3 par éclatements et enlèvements de diviseurs, qui est analogue aux constructions de Kapranov et Keel de M_0,n, respectivement à partir de P^n-3 et (P^1)^n-3.

Mots-clefs : Espaces de modules des courbes, théorie de Hodge mixte, opérades, valeurs zêta multiples, dualité de Koszul des opérades.

Abstract:
Francis Brown introduced a partial compactification M_0,n^ of the moduli space M_0,n. We prove that the gravity cooperad, given by the degree-shifted cohomologies of the spaces M_0,n, is cofree as a nonsymmetric anticyclic cooperad; moreover, the cogenerators are given by the cohomology groups of M_0,n^. As part of the proof we construct an explicit diagrammatically defined basis of H^(M_0,n) which is compatible with cooperadic cocomposition, and such that a subset forms a basis of H^(M_0,n^). We show that our results are equivalent to the claim that H^k(M _0,n^) has a pure Hodge structure of weight 2k for all k, and we conclude our paper by giving an independent and completely different proof of this fact. The latter proof uses a new and explicit iterative construction of M _0,n^ from A ^n-3 by blow-ups and removing divisors, analogous to Kapranov's and Keel's constructions of M _0,n from P ^n-3 and (P ^1)^n-3, respectively.

Keywords: Moduli of curves, mixed Hodge theory, operads, multiple zeta values, Koszul duality for operads.

Class. math. : 14H10, 11G55, 55P48, 18D50, 14F40, 11M32


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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