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},
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     year = {2017},
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     mrnumber = {3720025},
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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://archive.numdam.org/articles/10.24033/asens.2340/

Bibliographie
Cité par

Arbarello, E.; Cornalba, M. Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom., Volume 5 (1996), pp. 705-749 (ISSN: 1056-3911) | MR | Zbl

Alm, J. A universal A-infinity structure on Batalin-Vilkovisky algebras with multiple zeta value coefficients, Int. Math. Res. Not., Volume 2016 (2016), pp. 7414-7470 | DOI | MR | Zbl

Bergström, J.; Brown, F., Motives, quantum field theory, and pseudodifferential operators (Clay Math. Proc.), Volume 12, Amer. Math. Soc., Providence, RI, 2010, pp. 119-126 | MR | Zbl

Brown, F.; Carr, S.; Schneps, L. The algebra of cell-zeta values, Compos. Math., Volume 146 (2010), pp. 731-771 (ISSN: 0010-437X) | DOI | MR | Zbl

Brieskorn, E. Sur les groupes de tresses [d'après V. I. Arnol'd], Séminaire Bourbaki, vol. 1971/1972, exposé n^o 401, Lecture Notes in Math., Volume 317 (1973), pp. 21-44 | DOI | Numdam | MR | Zbl

Brown, F. C. S. Multiple zeta values and periods of moduli spaces ${\bar{𝔐}}_{0, n}$ , Ann. Sci. Éc. Norm. Supér., Volume 42 (2009), pp. 371-489 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Deligne, P. Théorie de Hodge. II, Publ. Math. IHÉS, Volume 40 (1971), pp. 5-57 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Deligne, P. La conjecture de Weil. II, Publ. Math. IHÉS, Volume 52 (1980), pp. 137-252 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Deligne, P.; Griffiths, P.; Morgan, J.; Sullivan, D. Real homotopy theory of Kähler manifolds, Invent. math., Volume 29 (1975), pp. 245-274 (ISSN: 0020-9910) | DOI | MR | Zbl

Dotsenko, V.; Shadrin, S.; Vallette, B. Givental group action on topological field theories and homotopy Batalin-Vilkovisky algebras, Adv. Math., Volume 236 (2013), pp. 224-256 (ISSN: 0001-8708) | DOI | MR | Zbl

Dupont, C.; Vallette, B. Brown's moduli spaces of curves and the gravity operad (preprint arXiv:1509.08840, to appear in Geometry and Topology ) | MR

Getzler, E., The moduli space of curves (Texel Island, 1994) (Progr. Math.), Volume 129, Birkhäuser, 1995, pp. 199-230 | DOI | MR | Zbl

Ginzburg, V.; Kapranov, M. Koszul duality for operads, Duke Math. J., Volume 76 (1994), pp. 203-272 (ISSN: 0012-7094) | DOI | MR | Zbl

Getzler, E.; Kapranov, M., Geometry, topology, & physics (Conf. Proc. Lecture Notes Geom. Topology, IV), Int. Press, Cambridge, MA, 1995, pp. 167-201 | MR | Zbl

Getzler, E.; Kapranov, M. Modular operads, Compos. math., Volume 110 (1998), pp. 65-126 (ISSN: 0010-437X) | DOI | MR | Zbl

Granåker, J. Strong homotopy properads, Int. Math. Res. Not., Volume 2007 (2007) | MR | Zbl

Guillén Santos, F.; Navarro, V.; Pascual, P.; Roig, A. Moduli spaces and formal operads, Duke Math. J., Volume 129 (2005), pp. 291-335 (ISSN: 0012-7094) | DOI | MR | Zbl

Gel'fand, I. M.; Zelevinskiĭ, A. V. Algebraic and combinatorial aspects of the general theory of hypergeometric functions, Funktsional. Anal. i Prilozhen., Volume 20 (1986), p. 17-34, 96 (ISSN: 0374-1990) | DOI | MR | Zbl

Hassett, B. Moduli spaces of weighted pointed stable curves, Adv. Math., Volume 173 (2003), pp. 316-352 (ISSN: 0001-8708) | DOI | MR | Zbl

Jambu, M.; Terao, H., Singularities (Iowa City, IA, 1986) (Contemp. Math.), Volume 90, Amer. Math. Soc., Providence, RI, 1989, pp. 147-162 | DOI | MR | Zbl

Kadeishvili, T. V. On the theory of homology of fiber spaces, Uspekhi Mat. Nauk, Volume 35 (1980), pp. 183-188 (ISSN: 0042-1316) | MR | Zbl

Kapranov, M., I. M. Gel'fand Seminar (Adv. Soviet Math.), Volume 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29-110 | DOI | MR | Zbl

Keel, S. Intersection theory of moduli space of stable $n$ -pointed curves of genus zero, Trans. Amer. Math. Soc., Volume 330 (1992), pp. 545-574 (ISSN: 0002-9947) | DOI | MR | Zbl

Li, L. Wonderful compactification of an arrangement of subvarieties, Michigan Math. J., Volume 58 (2009), pp. 535-563 (ISSN: 0026-2285) | DOI | MR | Zbl

Loday, J.-L.; Vallette, B., Grundl. math. Wiss., 346, Springer, Heidelberg, 2012, 634 pages (ISBN: 978-3-642-30361-6) | DOI | MR | Zbl

Menichi, L. Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, $K$ -Theory, Volume 32 (2004), pp. 231-251 (ISSN: 0920-3036) | DOI | MR | Zbl

Orlik, P.; Solomon, L. Combinatorics and topology of complements of hyperplanes, Invent. math., Volume 56 (1980), pp. 167-189 | DOI | MR | Zbl

Orlik, P.; Terao, H., Grundl. math. Wiss., 300, Springer, Berlin, 1992, 325 pages (ISBN: 3-540-55259-6) | DOI | MR | Zbl

Petersen, D. The structure of the tautological ring in genus one, Duke Math. J., Volume 163 (2014), pp. 777-793 (ISSN: 0012-7094) | DOI | MR | Zbl

Petersen, D. A spectral sequence for stratified spaces and configuration spaces of points, Geom. Topol., Volume 21 (2017), pp. 2527-2555 | DOI | MR | Zbl

Shapiro, B. Z. The mixed Hodge structure of the complement to an arbitrary arrangement of affine complex hyperplanes is pure, Proc. Amer. Math. Soc., Volume 117 (1993), pp. 931-933 (ISSN: 0002-9939) | DOI | MR | Zbl

Salvatore, P.; Tauraso, R. The operad Lie is free, J. Pure Appl. Algebra, Volume 213 (2009), pp. 224-230 | DOI | MR | Zbl

van der Laan, P. Operads up to homotopy and deformations of operad maps (preprint arXiv:math/0208041 )

Ward, B. C. Maurer-Cartan elements and cyclic operads, J. Noncommut. Geom., Volume 10 (2016), pp. 1403-1464 (ISSN: 1661-6952) | DOI | MR | Zbl

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