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

Brown, Francis C. S.

Multiple zeta values and periods of moduli spaces

{\bar{𝔐}}_{0, n}

[Valeurs zêta multiples et périodes des espaces de modules

{\bar{𝔐}}_{0, n}

]

Brown, Francis C. S.

Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009), p. 371-489 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

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.

Publié le : 2009-01-01

MR 2543329 | Zbl 1216.11079

DOI : https://doi.org/10.24033/asens.2099
Classification: 14G32, 11G55, 32G34
Mots clés: espace des modules, multizêtas, intégrales itérées, polylogarithmes, associateurs, associaèdres

@article{ASENS_2009_4_42_3_371_0,
     author = {Brown, Francis C. S.},
     title = {Multiple zeta values and periods of moduli spaces $\overline{\mathfrak {M}}\_{0,n}$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {42},
     year = {2009},
     pages = {371-489},
     doi = {10.24033/asens.2099},
     mrnumber = {2543329},
     zbl = {1216.11079},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2009_4_42_3_371_0}
}

Brown, Francis C. S. Multiple zeta values and periods of moduli spaces $\overline{\mathfrak {M}}_{0,n}$. Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009) pp. 371-489. doi : 10.24033/asens.2099. http://gdmltest.u-ga.fr/item/ASENS_2009_4_42_3_371_0/

Bibliographie

[1] K. Aomoto, Fonctions hyperlogarithmiques et groupes de monodromie unipotents, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978), 149-156. | MR 509582 | Zbl 0416.32020

[2] K. Aomoto, Addition theorem of Abel type for hyper-logarithms, Nagoya Math. J. 88 (1982), 55-71. | MR 683242 | Zbl 0545.33014

[3] K. Aomoto, Special values of hyperlogarithms and linear difference schemes, Illinois J. Math. 34 (1990), 191-216. | MR 1046562 | Zbl 0684.33010

[4] V. I. ArnolʼD, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227-231. | MR 242196 | Zbl 0277.55002

[5] A. A. Beilinson, A. B. Goncharov, V. V. Schechtman & A. N. Varchenko, Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of a pair of triangles in the plane, in Grothendieck Festschrift, 86, Birkhäuser, 1990, 135-171. | MR 1086885 | Zbl 0737.14003

[6] A. Borel & J-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436-491. | MR 387495 | Zbl 0274.22011

[7] F. C. S. Brown, Polylogarithmes multiples uniformes en une variable, C. R. Math. Acad. Sci. Paris 338 (2004), 527-532. | MR 2057024 | Zbl 1048.11053

[8] F. C. S. Brown, Single-valued hyperlogarithms and unipotent differential equations, preprint.

[9] K. T. Chen, Extension of $C^{\infty}$ function algebra by integrals and Malcev completion of $π_{1}$ , Advances in Math. 23 (1977), 181-210. | MR 458461 | Zbl 0345.58003

[10] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831-879. | MR 454968 | Zbl 0389.58001

[11] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math., Vol. 163, Springer, 1970. | Zbl 0244.14004

[12] P. Deligne, Théorie de Hodge. III, Publ. Math. I.H.É.S. 44 (1974), 5-77. | Numdam | MR 498552 | Zbl 0237.14003

[13] P. Deligne, Le groupe fondamental de la droite projective moins trois points 1987), Math. Sci. Res. Inst. Publ. 16, Springer, 1989, 79-297. | MR 1012168 | Zbl 0742.14022

[14] P. Deligne & A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup. 38 (2005), 1-56. | Numdam | MR 2136480 | Zbl 1084.14024

[15] S. L. Devadoss, Tessellations of moduli spaces and the mosaic operad, in Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math. 239, Amer. Math. Soc., 1999, 91-114. | MR 1718078 | Zbl 0968.32009

[16] S. L. Devadoss, Combinatorial equivalence of real moduli spaces, Notices Amer. Math. Soc. 51 (2004), 620-628. | MR 2064149 | Zbl 1093.14509

[17] A. C. Dixon, On a certain double integral, Proc. London Math. Soc. 2 (1905), 8-15. | JFM 35.0320.01 | MR 1577292

[18] V. G. DrinfelʼD, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal (\bar{𝐐} / 𝐐)$ , Leningrad Math. Journal 2 (1991), 829-860. | MR 1080203 | Zbl 0728.16021

[19] J. Écalle, Singularités non abordables par la géométrie, Ann. Inst. Fourier (Grenoble) 42 (1992), 73-164. | Numdam | MR 1162558 | Zbl 0940.32013

[20] M. Falk & R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), 77-88. | MR 808110 | Zbl 0574.55010

[21] S. Fischler, Groupes de Rhin-Viola et intégrales multiples, J. Théor. Nombres Bordeaux 15 (2003), 479-534. | Numdam | MR 2140865 | Zbl 1074.11040

[22] E. Getzler, Operads and moduli spaces of genus $0$ Riemann surfaces, in The moduli space of curves (Texel Island, 1994), Progr. Math. 129, Birkhäuser, 1995, 199-230. | MR 1363058 | Zbl 0851.18005

[23] A. B. Goncharov, The dihedral Lie algebras and Galois symmetries of $π_{1}^{(l)} (ℙ^{1} - ({0, \infty} \cup μ_{N}))$ , Duke Math. J. 110 (2001), 397-487. | MR 1869113 | Zbl 1113.14020

[24] A. B. Goncharov, Multiple $ζ$ -values, Galois groups, and geometry of modular varieties, in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, 2001, 361-392. | Zbl 1042.11042

[25] A. B. Goncharov, Multiple polylogarithms and mixed Tate motives, preprint arXiv:math.AG/0103059.

[26] A. B. Goncharov, Periods and mixed motives, preprint arXiv:math.AG/0202154.

[27] A. B. Goncharov & Y. I. Manin, Multiple $ζ$ -motives and moduli spaces ${\bar{ℳ}}_{0, n}$ , Compos. Math. 140 (2004), 1-14. | Zbl 1047.11063

[28] J. Gonzalez-Lorca, Série de Drinfel'd, monodromie et algèbres de Hecke, Thèse de doctorat, École Normale Supérieure, 1998.

[29] P. Griffiths & W. Schmid, Recent developments in Hodge theory: a discussion of techniques and results, in Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, 1975, 31-127. | MR 419850 | Zbl 0355.14003

[30] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publ. Math. I.H.É.S. 29 (1966), 95-103. | Numdam | MR 199194 | Zbl 0145.17602

[31] R. M. Hain, The geometry of the mixed Hodge structure on the fundamental group, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., 1987, 247-282. | MR 927984 | Zbl 0654.14006

[32] R. M. Hain, Classical polylogarithms, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., 1994, 3-42. | MR 1265550 | Zbl 0807.19003

[33] R. M. Hain & R. Macpherson, Higher logarithms, Illinois J. Math. 34 (1990), 392-475. | MR 1046570 | Zbl 0737.14014

[34] M. Hoang Ngoc, M. Petitot & J. Van Den Hoeven, Shuffle algebra and polylogarithms, in Formal Power Series and Algebraic Combinatorics 1998, Toronto.

[35] M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin. 11 (2000), 49-68. | MR 1747062 | Zbl 0959.16021

[36] M. M. Kapranov, The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation, J. Pure Appl. Algebra 85 (1993), 119-142. | MR 1207505 | Zbl 0812.18003

[37] V. G. Knizhnik & A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), 83-103. | MR 853258 | Zbl 0661.17020

[38] T. Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985), 57-75. | MR 808109 | Zbl 0574.55009

[39] E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, 1973, Pure and Applied Mathematics, Vol. 54. | MR 568864 | Zbl 0264.12102

[40] I. Lappo-Danilevskii, Mémoires sur la théorie des systèmes des équations différentielles linéaires, Chelsea, New York, 1953. | Zbl 0011.34903

[41] A. R. Magid, Lectures on differential Galois theory, University Lecture Series 7, Amer. Math. Soc., 1994. | MR 1301076 | Zbl 0855.12001

[42] L. Boutet De Monvel, Polylogarithmes, http://www.math.jussieu.fr/~boutet.

[43] P. Orlik & H. Terao, Arrangements of hyperplanes, Grund. Math. Wiss. 300, Springer, 1992. | MR 1217488 | Zbl 0757.55001

[44] H. Poincaré, Sur les groupes d'équations linéaires, Acta Mathematica 4 (1884). | JFM 16.0252.01

[45] G. Racinet, Doubles mélanges des polylogarithmes multiples aux racines de l'unité, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 185-231. | Numdam | MR 1953193 | Zbl 1050.11066

[46] D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra 58 (1979), 432-454. | MR 540649 | Zbl 0409.16011

[47] G. Rhin & C. Viola, On a permutation group related to $ζ (2)$ , Acta Arith. 77 (1996), 23-56. | MR 1404975 | Zbl 0864.11037

[48] J. D. Stasheff, Homotopy associativity of $H$ -spaces. I, Trans. Amer. Math. Soc. 108 (1963), 275-292. | Zbl 0114.39402

[49] T. Terasoma, Selberg integrals and multiple zeta values, Compositio Math. 133 (2002), 1-24. | MR 1918286 | Zbl 1003.11042

[50] C. Voisin, Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics 77, Cambridge University Press, 2003. | MR 1997577 | Zbl 1032.14002

[51] M. Waldschmidt, Valeurs zêta multiples. Une introduction, J. Théor. Nombres Bordeaux 12 (2000), 581-595. | Numdam | MR 1823204 | Zbl 0976.11037

[52] M. Yoshida, Fuchsian differential equations, Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, 1987. | MR 986252 | Zbl 0618.35001

[53] J. Zhao, Multiple polylogarithms: analytic continuation, monodromy, and variations of mixed Hodge structures, in Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000), Nankai Tracts Math. 5, World Sci. Publ., River Edge, NJ, 2002, 167-193. | MR 1945360 | Zbl 1058.32005

[54] S. A. Zlobin, Integrals that can be represented as linear forms of generalized polylogarithms, Mat. Zametki [Math. Notes] 71 (2002), 782-787 [711-716]. | MR 1936201 | Zbl 1049.11077

[55] W. Zudilin, Well-poised hypergeometric transformations of Euler-type multiple integrals, J. London Math. Soc. 70 (2004), 215-230. | MR 2064759 | Zbl 1065.11054