Brown's dihedral moduli space and freedom of the gravity operad

Alm, Johan; Petersen, Dan

doi:10.24033/asens.2340

Brown's dihedral moduli space and freedom of the gravity operad
[Espace de modules dièdre de Brown et liberté de l'opérade de gravité]

Alm, Johan ; Petersen, Dan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 5, pp. 1081-1122.

Résumé
Abstract

Francis Brown a introduit une compactification partielle $M_{0, n}^{δ}$ 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 $M_{0, n}^{δ}$ . 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 $H^{•} (M_{0, n}^{δ})$ . Nous montrons que nos résultats sont équivalents au fait que $H^{k} (M_{0, n}^{δ})$ a une structure de Hodge pure de poids $2 k$ 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 $M_{0, n}^{δ}$ à partir de $𝔸^{n - 3}$ par éclatements et enlèvements de diviseurs, qui est analogue aux constructions de Kapranov et Keel de ${\bar{M}}_{0, n}$ , respectivement à partir de $ℙ^{n - 3}$ et ${(ℙ^{1})}^{n - 3}$ .

Francis Brown introduced a partial compactification $M_{0, n}^{δ}$ 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 $M_{0, n}^{δ}$ . 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 $H^{•} (M_{0, n}^{δ})$ . We show that our results are equivalent to the claim that $H^{k} (M_{0, n}^{δ})$ has a pure Hodge structure of weight $2 k$ 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 $M_{0, n}^{δ}$ from $𝔸^{n - 3}$ by blow-ups and removing divisors, analogous to Kapranov's and Keel's constructions of ${\bar{M}}_{0, n}$ from $¶^{n - 3}$ and ${(¶^{1})}^{n - 3}$ , respectively.

MR Zbl

DOI : 10.24033/asens.2340

Classification : 14H10, 11G55, 55P48, 18D50, 14F40, 11M32
Keywords: Moduli of curves, mixed Hodge theory, operads, multiple zeta values, Koszul duality for operads.
Mot clés : Espaces de modules des courbes, théorie de Hodge mixte, opérades, valeurs zêta multiples, dualité de Koszul des opérades.

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     author = {Alm, Johan and Petersen, Dan},
     title = {Brown's dihedral moduli space and freedom of the gravity operad},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1081--1122},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {5},
     year = {2017},
     doi = {10.24033/asens.2340},
     mrnumber = {3720025},
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     url = {http://www.numdam.org/articles/10.24033/asens.2340/}
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Alm, Johan; Petersen, Dan. Brown's dihedral moduli space and freedom of the gravity operad. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 5, pp. 1081-1122. doi : 10.24033/asens.2340. http://www.numdam.org/articles/10.24033/asens.2340/

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