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.
@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},
pages = {253--268},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 10},
number = {2},
year = {2011},
mrnumber = {2856148},
zbl = {1239.32023},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2011_5_10_2_253_0/}
}
TY - JOUR AU - Dimca, Alexandru AU - Papadima, Stefan TI - Finite Galois covers, cohomology jump loci, formality properties, and multinets JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 253 EP - 268 VL - 10 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2011_5_10_2_253_0/ LA - en ID - ASNSP_2011_5_10_2_253_0 ER -
%0 Journal Article %A Dimca, Alexandru %A Papadima, Stefan %T Finite Galois covers, cohomology jump loci, formality properties, and multinets %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 253-268 %V 10 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2011_5_10_2_253_0/ %G en %F 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, Série 5, Tome 10 (2011) no. 2, pp. 253-268. http://archive.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 | Zbl
[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 | Zbl
[3] B. Berceau and S. Papadima, Universal representations of braid and braid-permutation groups, J. Knot Theory Ramifications 18 (2009), 999–1019. | MR | Zbl
[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 | Zbl
[5] E. Brieskorn, Sur les groupes de tresses, In: “Séminaire Bourbaki”, 1971/72, Lect. Notes in Math. Vol. 317, Springer-Verlag, 1973, 21–44. | EuDML | Numdam | MR | Zbl
[6] K. S. Brown, “Cohomology of Groups”, Grad. Texts in Math., Vol. 87, Springer-Verlag, New York-Berlin, 1982. | MR | Zbl
[7] N. Budur, A. Dimca and M. Saito, First Milnor cohomology of hyperplane arrangements, Contemp. Math. 538 (2011), 279–292. | MR | Zbl
[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. | EuDML | MR | Zbl
[9] D. C. Cohen and A. I. Suciu, On Milnor fibrations of arrangements, J. London Math. Soc. 51 (1995), 105–119. | MR | Zbl
[10] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–274. | EuDML | MR | Zbl
[11] A. Dimca, “Singularities and Topology of Hypersurfaces”, Universitext, Springer-Verlag, 1992. | MR | Zbl
[12] A. Dimca, “Sheaves in Topology”, Universitext, Springer-Verlag, 2004. | MR | Zbl
[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 | Zbl
[14] A. Dimca, On the isotropic subspace theorems, Bull. Math. Soc. Sci. Math. Roumanie 51 (2008), 307–324. | MR | Zbl
[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 | Zbl
[16] A. Dimca, On admissible rank one local systems, J. Algebra 321 (2009), 3145–3157. | MR | Zbl
[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 | Zbl
[18] A. Dimca, S. Papadima and A. Suciu, Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009), 405–457. | MR | Zbl
[19] M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143 (2007), 1069–1088. | MR | Zbl
[20] M. Green and R. Lazarfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4 (1991), 87–103. | MR | Zbl
[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. | EuDML | Numdam | MR | Zbl
[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 | Zbl
[23] A. Macinic, Cohomology rings and formality properties of nilpotent groups, J. Pure Appl. Algebra 214 (2010), 1818–1826. | MR | Zbl
[24] A. Macinic and S. Papadima, On the monodromy action on Milnor fibers of graphic arrangements, Topology Appl. 156 (2009), 761–774. | MR | Zbl
[25] J. W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204. | EuDML | Numdam | MR | Zbl
[26] P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189. | EuDML | MR | Zbl
[27] S. Papadima and A. I. Suciu, Toric complexes and Artin kernels, Adv. Math. 220 (2009), 441–477. | MR | Zbl
[28] S. Papadima and A. I. Suciu, Algebraic monodromy and obstructions to formality, Forum Math. 22 (2010), 973–983. | MR | Zbl
[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 | Zbl
[30] C. Peters and J. Steenbrink, “Mixed Hodge Structures”, Ergeb. der Math. und ihrer Grenz. 3, Folge 52, Springer, 2008. | MR | Zbl
[31] D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205–295. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[35] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331. | EuDML | Numdam | MR | Zbl
[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 | Zbl
[37] H. Zuber, Non-formality of Milnor fibres of line arrangements, Bull. London Math. Soc. 42 (2010), 905–911. | MR | Zbl
