Finite Galois covers, cohomology jump loci, formality properties, and multinets
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, p. 253-268

We explore the relation between cohomology jump loci in a finite Galois cover, formality properties and algebraic monodromy action. We show that the jump loci of the base and total space are essentially the same, provided the base space is 1-formal and the monodromy action in degree 1 is trivial. We use reduced multinet structures on line arrangements to construct components of the first characteristic variety of the Milnor fiber in degree 1, and to prove that the monodromy action is non-trivial in degree 1. For an arbitrary line arrangement, we prove that the triviality of the monodromy in degree 1 can be detected in a precise way, by resonance varieties.

Published online : 2018-08-07
Classification:  32S22,  52C30,  55N25,  55P62
@article{ASNSP_2011_5_10_2_253_0,
     author = {Dimca, Alexandru and Papadima, Stefan},
     title = {Finite Galois covers, cohomology jump loci, formality properties, and multinets},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {2},
     year = {2011},
     pages = {253-268},
     zbl = {1239.32023},
     mrnumber = {2856148},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_253_0}
}
Dimca, Alexandru; Papadima, Stefan. Finite Galois covers, cohomology jump loci, formality properties, and multinets. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 253-268. http://www.numdam.org/item/ASNSP_2011_5_10_2_253_0/

[1] D. Arapura, Geometry of cohomology support loci for local systems. I, J. Algebraic Geom. 6 (1997), 563–597. | MR 1487227 | Zbl 0923.14010

[2] A. Beauville, Annulation du H1 pour les fibrés en droites plats, In: “Complex Algebraic Varieties” (Bayreuth, 1990), Lect. Notes in Math., Vol. 1507, Springer, Berlin, 1992, 1–15. | MR 1178716 | Zbl 0792.14006

[3] B. Berceau and S. Papadima, Universal representations of braid and braid-permutation groups, J. Knot Theory Ramifications 18 (2009), 999–1019. | MR 2549480 | Zbl 1220.20029

[4] A. K. Bousfiled and V. K. A. M. Gugenheim, “On PL De Rham Theory and Rational Homotopy Type”, Memoirs Amer. Math. Soc., Vol. 179, Amer. Math. Soc., Providence, RI, 1976. | MR 425956 | Zbl 0338.55008

[5] E. Brieskorn, Sur les groupes de tresses, In: “Séminaire Bourbaki”, 1971/72, Lect. Notes in Math. Vol. 317, Springer-Verlag, 1973, 21–44. | Numdam | MR 347513 | Zbl 0277.55003

[6] K. S. Brown, “Cohomology of Groups”, Grad. Texts in Math., Vol. 87, Springer-Verlag, New York-Berlin, 1982. | MR 672956 | Zbl 0584.20036

[7] N. Budur, A. Dimca and M. Saito, First Milnor cohomology of hyperplane arrangements, Contemp. Math. 538 (2011), 279–292. | MR 2777825 | Zbl 1228.32027

[8] A. D. R. Choudary, A. Dimca and S. Papadima, Some analogs of Zariski’s theorem on nodal line arrangements, Algebr. Geom. Topol. 5 (2005), 691–711. | MR 2153112 | Zbl 1081.32018

[9] D. C. Cohen and A. I. Suciu, On Milnor fibrations of arrangements, J. London Math. Soc. 51 (1995), 105–119. | MR 1310725 | Zbl 0814.32007

[10] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–274. | MR 382702 | Zbl 0312.55011

[11] A. Dimca, “Singularities and Topology of Hypersurfaces”, Universitext, Springer-Verlag, 1992. | MR 1194180 | Zbl 0753.57001

[12] A. Dimca, “Sheaves in Topology”, Universitext, Springer-Verlag, 2004. | MR 2050072 | Zbl 1043.14003

[13] A. Dimca, Characteristic varieties and constructible sheaves, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007), 365–389. | MR 2349994 | Zbl 1139.14009

[14] A. Dimca, On the isotropic subspace theorems, Bull. Math. Soc. Sci. Math. Roumanie 51 (2008), 307–324. | MR 2447175 | Zbl 1174.14008

[15] A. Dimca, Pencils of plane curves and characteristic varieties, In: “Arrangements, Local Systems and Singularities”, Progress in Mathematics, Vol. 283, Birkhäuser, 2009, 59–82. | MR 3025860 | Zbl 1288.14005

[16] A. Dimca, On admissible rank one local systems, J. Algebra 321 (2009), 3145–3157. | MR 2510044 | Zbl 1186.14018

[17] A. Dimca, S. Papadima and A. Suciu, Quasi-Kähler groups, 3-manifold groups, and formality, Math. Z., available online DOI: 10.1007/s00209-010-0664-y | MR 2805428 | Zbl 1228.14018

[18] A. Dimca, S. Papadima and A. Suciu, Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009), 405–457. | MR 2527322 | Zbl 1222.14035

[19] M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143 (2007), 1069–1088. | MR 2339840 | Zbl 1122.52009

[20] M. Green and R. Lazarfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4 (1991), 87–103. | MR 1076513 | Zbl 0735.14004

[21] M. Kapovich and J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 5–95. | Numdam | MR 1733326 | Zbl 0982.20023

[22] A. Libgober, Eigenvalues for the monodromy of the Milnor fibers of arrangements, In: “Trends in singularities”, Trends Math., Birkhäuser, Basel, 2002, 141–150. | MR 1900784 | Zbl 1036.32019

[23] A. Macinic, Cohomology rings and formality properties of nilpotent groups, J. Pure Appl. Algebra 214 (2010), 1818–1826. | MR 2608110 | Zbl 1238.20052

[24] A. Macinic and S. Papadima, On the monodromy action on Milnor fibers of graphic arrangements, Topology Appl. 156 (2009), 761–774. | MR 2492960 | Zbl 1170.32009

[25] J. W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204. | Numdam | MR 516917 | Zbl 0401.14003

[26] P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189. | MR 558866 | Zbl 0432.14016

[27] S. Papadima and A. I. Suciu, Toric complexes and Artin kernels, Adv. Math. 220 (2009), 441–477. | MR 2466422 | Zbl 1208.57002

[28] S. Papadima and A. I. Suciu, Algebraic monodromy and obstructions to formality, Forum Math. 22 (2010), 973–983. | MR 2719766 | Zbl 1229.57002

[29] S. Papadima and A. I. Suciu, Geometric and algebraic aspects of 1-formality, Bull. Math. Soc. Sci. Math. Roumanie 52 (2009), 355–375. | MR 2554494 | Zbl 1199.55010

[30] C. Peters and J. Steenbrink, “Mixed Hodge Structures”, Ergeb. der Math. und ihrer Grenz. 3, Folge 52, Springer, 2008. | MR 2393625 | Zbl 1138.14002

[31] D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205–295. | MR 258031 | Zbl 0191.53702

[32] B. Z. Shapiro, The mixed Hodge structure of the complement to an arbitrary arrangement of affine complex hyperplanes is pure, Proc. Amer. Math. Soc. 117 (1993), 931–933. | MR 1131042 | Zbl 0798.32029

[33] J. Steenbrink, Mixed Hodge structures associated with isolated singularities, In: “Singularities”, Part 2 (Arcata, 1981), Proc. Symp. Pure Math. 40, Amer. Math. Soc., 1983, 513–536. | MR 713277 | Zbl 0515.14003

[34] A. Suciu, Fundamental groups of line arrangements: enumerative aspects, In: “Advances in Algebraic Geometry Motivated by Physics” (Lowell, MA, 2000), Contemp. Math. Vol. 276, Amer. Math. Soc., Providence, RI, 2001, 43–79. | MR 1837109 | Zbl 0998.14012

[35] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331. | Numdam | MR 646078 | Zbl 0374.57002

[36] S. Yuzvinsky, A new bound on the number of special fibers in a pencil of curves, Proc. Amer. Math. Soc. 137 (2009), 1641–1648. | MR 2470822 | Zbl 1173.14021

[37] H. Zuber, Non-formality of Milnor fibres of line arrangements, Bull. London Math. Soc. 42 (2010), 905–911. | MR 2728693 | Zbl 1202.32022