Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Relative cohomology of bi-arrangements

Author: Clément Dupont
Journal: Trans. Amer. Math. Soc. 369 (2017), 8105-8160
MSC (2010): Primary 14C30, 14F05, 14F25, 52C35
Published electronically: May 30, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A bi-arrangement of hyperplanes in a complex affine space is the data of two sets of hyperplanes along with coloring information on the strata. To such a bi-arrangement one naturally associates a relative cohomology group that we call its motive. The main reason for studying such relative cohomology groups comes from the notion of motivic period. More generally, we suggest the systematic study of the motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute the structure of these motives. Our main object is the Orlik-Solomon bi-complex of a bi-arrangement, which generalizes the Orlik-Solomon algebra of an arrangement. Loosely speaking, our main result states that the motive of an exact bi-arrangement is computed by its Orlik-Solomon bi-complex, which generalizes classical facts involving the Orlik-Solomon algebra of an arrangement. We show how this formalism allows us to explicitly compute motives arising from the study of multiple zeta values and sketch a more general application to periods of mixed Tate motives.

References [Enhancements On Off] (What's this?)

  • [AM09] Paolo Aluffi and Matilde Marcolli, Feynman motives of banana graphs, Commun. Number Theory Phys. 3 (2009), no. 1, 1-57. MR 2504753,
  • [And09] Yves André, Galois theory, motives and transcendental numbers, Renormalization and Galois theories, IRMA Lect. Math. Theor. Phys., vol. 15, Eur. Math. Soc., Zürich, 2009, pp. 165-177. MR 2588609,
  • [Apé79] R. Apéry, Irrationalité de $ \zeta (2)$ et $ \zeta (3)$., Astérisque 61 (1979), 11-13 (French).
  • [Arn69] V. I. Arnold, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227-231 (Russian). MR 0242196
  • [BEK06] Spencer Bloch, Hélène Esnault, and Dirk Kreimer, On motives associated to graph polynomials, Comm. Math. Phys. 267 (2006), no. 1, 181-225. MR 2238909,
  • [Blo12] S. Bloch, Motives, the fundamental group, and graphs, preprint, 2012.
  • [BR01] Keith Ball and Tanguy Rivoal, Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math. 146 (2001), no. 1, 193-207 (French). MR 1859021,
  • [Bri73] Egbert Brieskorn, Sur les groupes de tresses [d'après V. I. Arnold], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Math., vol. 317, Springer, Berlin, 1973, pp. 21-44 (French). MR 0422674
  • [Bro09] Francis C. S. Brown, Multiple zeta values and periods of moduli spaces $ \overline {\mathfrak{M}}_{0,n}$, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 3, 371-489 (English, with English and French summaries). MR 2543329
  • [Bro12] Francis Brown, Mixed Tate motives over $ \mathbb{Z}$, Ann. of Math. (2) 175 (2012), no. 2, 949-976. MR 2993755,
  • [BS12] Francis Brown and Oliver Schnetz, A K3 in $ \phi ^4$, Duke Math. J. 161 (2012), no. 10, 1817-1862. MR 2954618,
  • [BVGS90] A. A. Beĭlinson, A. N. Varchenko, A. B. Goncharov, and V. V. Shekhtman, Projective geometry and $ K$-theory, Algebra i Analiz 2 (1990), no. 3, 78-130 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 3, 523-576. MR 1073210
  • [Del71] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-57 (French). MR 0498551
  • [Del74] Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5-77. MR 0498552 (58:16653b)
  • [Del89] P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over $ {\bf Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 79-297 (French). MR 1012168,
  • [DG05] Pierre Deligne and Alexander B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 1, 1-56 (French, with English and French summaries). MR 2136480,
  • [Dor10] Dzmitry Doryn, Cohomology of graph hypersurfaces associated to certain Feynman graphs, Commun. Number Theory Phys. 4 (2010), no. 2, 365-415. MR 2725055,
  • [Dup14a] Clément Dupont, The combinatorial Hopf algebra of motivic dissection polylogarithms, Adv. Math. 264 (2014), 646-699. MR 3250295,
  • [Dup14b] Clément Dupont, Periods of hyperplane arrangements and motivic coproduct, PhD Thesis, Université Paris 6 Pierre et Marie Curie (2014).
  • [Dup15] Clément Dupont, The Orlik-Solomon model for hypersurface arrangements, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 6, 2507-2545 (English, with English and French summaries). MR 3449588
  • [GM04] A. B. Goncharov and Yu. I. Manin, Multiple $ \zeta $-motives and moduli spaces $ \overline {\mathcal {M}}_{0,n}$, Compos. Math. 140 (2004), no. 1, 1-14. MR 2004120,
  • [Gon02] A. B. Goncharov, Periods and mixed motives, preprint, arXiv:math/0202154, 2002.
  • [Gon05] A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005), no. 2, 209-284. MR 2140264,
  • [HMS11] A. Huber and S. Müller-Stach, On the relation between Nori motives and Kontsevich periods, preprint, 2011.
  • [Hub00] Annette Huber, Realization of Voevodsky's motives, J. Algebraic Geom. 9 (2000), no. 4, 755-799. MR 1775312
  • [Hub04] A. Huber, Corrigendum to: ``Realization of Voevodsky's motives'' [J. Algebraic Geom. 9 (2000), no. 4, 755-799; MR1775312], J. Algebraic Geom. 13 (2004), no. 1, 195-207. MR 2008720,
  • [Kon99] Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35-72. MR 1718044,
  • [KZ01] Maxim Kontsevich and Don Zagier, Periods, Mathematics unlimited--2001 and beyond, Springer, Berlin, 2001, pp. 771-808. MR 1852188
  • [Lev93] Marc Levine, Tate motives and the vanishing conjectures for algebraic $ K$-theory, Algebraic $ K$-theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 167-188. MR 1367296
  • [Li09] Li Li, Wonderful compactification of an arrangement of subvarieties, Michigan Math. J. 58 (2009), no. 2, 535-563. MR 2595553,
  • [Loo93] Eduard Looijenga, Cohomology of $ {\mathcal {M}}_3$ and $ {\mathcal {M}}^1_3$, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 205-228. MR 1234266,
  • [Mar10] Matilde Marcolli, Feynman integrals and motives, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 293-332. MR 2648331,
  • [MSWZ12] Stefan Müller-Stach, Stefan Weinzierl, and Raphael Zayadeh, A second-order differential equation for the two-loop sunrise graph with arbitrary masses, Commun. Number Theory Phys. 6 (2012), no. 1, 203-222. MR 2955935,
  • [OS80] Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167-189. MR 558866,
  • [OT92] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488
  • [Oxl11] James Oxley, Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, Oxford, 2011. MR 2849819
  • [PS08] Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 52, Springer-Verlag, Berlin, 2008. MR 2393625
  • [Ter02] Tomohide Terasoma, Mixed Tate motives and multiple zeta values, Invent. Math. 149 (2002), no. 2, 339-369. MR 1918675,
  • [Zha04] Jianqiang Zhao, Motivic cohomology of pairs of simplices, Proc. London Math. Soc. (3) 88 (2004), no. 2, 313-354. MR 2032510,
  • [Zud01] V. V. Zudilin, One of the numbers $ \zeta (5)$, $ \zeta (7)$, $ \zeta (9)$, $ \zeta (11)$ is irrational, Uspekhi Mat. Nauk 56 (2001), no. 4(340), 149-150 (Russian); English transl., Russian Math. Surveys 56 (2001), no. 4, 774-776. MR 1861452,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14C30, 14F05, 14F25, 52C35

Retrieve articles in all journals with MSC (2010): 14C30, 14F05, 14F25, 52C35

Additional Information

Clément Dupont
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Address at time of publication: Institut Montpelliérain Alexander Grothendieck, CNRS, Université Montpellier, France

Keywords: Arrangements, relative cohomology, mixed Hodge structures
Received by editor(s): March 12, 2015
Received by editor(s) in revised form: January 18, 2016, and January 19, 2016
Published electronically: May 30, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society