Orlik-Solomon type presentations for the cohomology algebra of toric arrangements

Callegaro, Filippo; D’Adderio, Michele; Delucchi, Emanuele; Migliorini, Luca; Pagaria, Roberto

doi:10.1090/tran/7952

Orlik-Solomon type presentations for the cohomology algebra of toric arrangements
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by Filippo Callegaro, Michele D’Adderio, Emanuele Delucchi, Luca Migliorini and Roberto Pagaria PDF

Trans. Amer. Math. Soc. 373 (2020), 1909-1940 Request permission

Corrigendum: Trans. Amer. Math. Soc. 374 (2021), 3779-3781.

Abstract:

We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias “toric arrangement”). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes and builds on De Concini and Procesi’s work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont’s rational formality result to formality over $\mathbb Z$.

The data needed in order to state the presentation of the rational cohomology is fully encoded in the poset of connected components of intersections of the arrangement.

References

V. I. Arnol′d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227–231 (Russian). MR 242196
Christin Bibby, Cohomology of abelian arrangements, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3093–3104. MR 3487239, DOI 10.1090/proc/12937
Christin Bibby, Representation stability for the cohomology of arrangements associated to root systems, J. Algebraic Combin. 48 (2018), no. 1, 51–75. MR 3836246, DOI 10.1007/s10801-017-0792-0
Petter Brändén and Luca Moci, The multivariate arithmetic Tutte polynomial, Trans. Amer. Math. Soc. 366 (2014), no. 10, 5523–5540. MR 3240933, DOI 10.1090/S0002-9947-2014-06092-3
Egbert Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol′d], Séminaire Bourbaki, 24ème année (1971/1972), Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. Exp. No. 401, pp. 21–44 (French). MR 0422674
Filippo Callegaro and Emanuele Delucchi, The integer cohomology algebra of toric arrangements, Adv. Math. 313 (2017), 746–802. MR 3649237, DOI 10.1016/j.aim.2017.04.017
Filippo Callegaro and Emanuele Delucchi, Erratum to: “The integer cohomology algebra of toric arrangements”, arXiv:1910.13836 (2019).
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013113
Giacomo d’Antonio and Emanuele Delucchi, A Salvetti complex for toric arrangements and its fundamental group, Int. Math. Res. Not. IMRN 15 (2012), 3535–3566. MR 2959041, DOI 10.1093/imrn/rnr161
Giacomo d’Antonio and Emanuele Delucchi, Minimality of toric arrangements, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 3, 483–521. MR 3323196, DOI 10.4171/JEMS/508
Corrado De Concini and Giovanni Gaiffi, Projective wonderful models for toric arrangements, Adv. Math. 327 (2018), 390–409. MR 3761997, DOI 10.1016/j.aim.2017.06.019
Corrado De Concini and Giovanni Gaiffi, Cohomology rings of compactifications of toric arrangements, Algebr. Geom. Topol. 19 (2019), no. 1, 503–532. MR 3910589, DOI 10.2140/agt.2019.19.503
Michele D’Adderio and Luca Moci, Arithmetic matroids, the Tutte polynomial and toric arrangements, Adv. Math. 232 (2013), 335–367. MR 2989987, DOI 10.1016/j.aim.2012.09.001
C. De Concini and C. Procesi, On the geometry of toric arrangements, Transform. Groups 10 (2005), no. 3-4, 387–422. MR 2183118, DOI 10.1007/s00031-005-0403-3
Corrado De Concini and Claudio Procesi, Topics in hyperplane arrangements, polytopes and box-splines, Universitext, Springer, New York, 2011. MR 2722776
C. De Concini, C. Procesi, and M. Vergne, Vector partition functions and index of transversally elliptic operators, Transform. Groups 15 (2010), no. 4, 775–811. MR 2753257, DOI 10.1007/s00031-010-9101-x
Emanuele Delucchi and Sonja Riedel, Group actions on semimatroids, Adv. in Appl. Math. 95 (2018), 199–270. MR 3759217, DOI 10.1016/j.aam.2017.11.001
Graham Denham, Alexander I. Suciu, and Sergey Yuzvinsky, Abelian duality and propagation of resonance, Selecta Math. (N.S.) 23 (2017), no. 4, 2331–2367. MR 3703455, DOI 10.1007/s00029-017-0343-5
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
Clément Dupont, Purity, formality, and arrangement complements, Int. Math. Res. Not. IMRN 13 (2016), 4132–4144. MR 3544631, DOI 10.1093/imrn/rnv260
Alex Fink and Luca Moci, Matroids over a ring, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 4, 681–731. MR 3474454, DOI 10.4171/JEMS/600
A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95–103. MR 199194
Jim Lawrence, Characteristic polynomials, Ehrhart quasi-polynomials, and torus groups, J. Number Theory 117 (2006), no. 2, 315–329. MR 2213768, DOI 10.1016/j.jnt.2005.06.012
Jim Lawrence, Enumeration in torus arrangements, European J. Combin. 32 (2011), no. 6, 870–881. MR 2821558, DOI 10.1016/j.ejc.2011.02.003
Eduard Looijenga, Cohomology of ${\scr M}_3$ and ${\scr M}^1_3$, 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, DOI 10.1090/conm/150/01292
Luca Moci, A Tutte polynomial for toric arrangements, Trans. Amer. Math. Soc. 364 (2012), no. 2, 1067–1088. MR 2846363, DOI 10.1090/S0002-9947-2011-05491-7
Luca Moci, Wonderful models for toric arrangements, Int. Math. Res. Not. IMRN 1 (2012), 213–238. MR 2874932, DOI 10.1093/imrn/rnr016
Luca Moci and Simona Settepanella, The homotopy type of toric arrangements, J. Pure Appl. Algebra 215 (2011), no. 8, 1980–1989. MR 2776437, DOI 10.1016/j.jpaa.2010.11.008
Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167–189. MR 558866, DOI 10.1007/BF01392549
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, DOI 10.1007/978-3-662-02772-1
James Oxley, Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, Oxford, 2011. MR 2849819, DOI 10.1093/acprof:oso/9780198566946.001.0001
Roberto Pagaria, Combinatorics of toric arrangements, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. vol. 30, Issue 2, 2019, pp. 317-349, DOI 10.4171/RLM/849
Roberto Pagaria, Two examples of toric arrangements, J. Combin. Theory Ser. A 167 (2019), 389–402. MR 3954076, DOI 10.1016/j.jcta.2019.05.006
S. Yuzvinskiĭ, Orlik-Solomon algebras in algebra and topology, Uspekhi Mat. Nauk 56 (2001), no. 2(338), 87–166 (Russian, with Russian summary); English transl., Russian Math. Surveys 56 (2001), no. 2, 293–364. MR 1859708, DOI 10.1070/RM2001v056n02ABEH000383
Thomas Zaslavsky, A combinatorial analysis of topological dissections, Advances in Math. 25 (1977), no. 3, 267–285. MR 446994, DOI 10.1016/0001-8708(77)90076-7

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Orlik-Solomon type presentations for the cohomology algebra of toric arrangements
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Abstract: