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ON THE MILNOR MONODROMY OF THE IRREDUCIBLE COMPLEX REFLECTION ARRANGEMENTS

Part of: Singularities

Published online by Cambridge University Press:  08 November 2017

Alexandru Dimca Open the ORCID record for Alexandru Dimca [Opens in a new window]
Alexandru Dimca*
Affiliation:
Université Côte d’Azur, CNRS, LJAD, France ([email protected])
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Abstract

Using recent results by Măcinic, Papadima and Popescu, and a refinement of an older construction of ours, we determine the monodromy action on $H^{1}(F(G),\mathbb{C})$, where $F(G)$ denotes the Milnor fiber of a hyperplane arrangement associated to an irreducible complex reflection group $G$.

Keywords

plane curvesMilnor fibermonodromycomplex reflection group

MSC classification

Primary: 32S55: Milnor fibration; relations with knot theory
Secondary: 32S35: Mixed Hodge theory of singular varieties 32S22: Relations with arrangements of hyperplanes
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 6 , November 2019 , pp. 1215 - 1231
Copyright
© Cambridge University Press 2017 

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Footnotes

The author was partially supported by Institut Universitaire de France.

References

Bailet, P., Dimca, A. and Yoshinaga, M., A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements, preprint, 2017,arXiv:1705.06022.Google Scholar
Bailet, P. and Settepanella, S., Homology graph of real arrangements and monodromy of Milnor fiber, Adv. Appl. Math. 90 (2017), 46–85.Google Scholar
Beltrametti, M. and Sommese, A.J., On k-jet ampleness, in Complex Analysis and Geometry (ed. Silva, Ancona), pp. 355–376 (Plenum Press, NY, 1993).Google Scholar
Bessis, D., Finite complex reflection arrangements are K (𝜋, 1), Ann. of Math. (2) 181(2015) 809–904.Google Scholar
Budur, N., Dimca, A. and Saito, M., First Milnor cohomology of hyperplane arrangements, Contemp. Math. 538 (2011), 279–292.Google Scholar
Cohen, D. C. and Suciu, A. I., On Milnor fibrations of arrangements, J. Lond. Math. Soc. 51(2) (1995), 105–119.Google Scholar
Deligne, P. and Dimca, A., Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulières, Ann. Sci. Éc. Norm. Supér. (4) 23 (1990), 645–656.Google Scholar
Dimca, A., Betti numbers of hypersurfaces and defects of linear systems, Duke Math. J. 60 (1990), 285–298.Google Scholar
Dimca, A., Singularities and Topology of Hypersurfaces, Universitext (Springer, New York, 1992).Google Scholar
Dimca, A., Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements, Nagoya Math. J. 206 (2012), 75–97.Google Scholar
Dimca, A., Hyperplane Arrangements: An Introduction, Universitext (Springer, Cham, Switzerland, 2017).Google Scholar
Dimca, A. and Lehrer, G., Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements, in Configuration Spaces, Geometry, Combinatorics and Topology, Publications of Scuola Normale Superiore, Volume 14, pp. 231–253 (Edizioni della Normale, Pisa, Italy, 2012).Google Scholar
Dimca, A. and Lehrer, G., Cohomology of the Milnor fiber of a hyperplane arrangement with symmetry, in Configuration Spaces – Geometry, Topology and Representation Theory, Cortona 2014 (Springer, Cham, Switzerland, 2016).Google Scholar
Dimca, A., Saito, M. and Wotzlaw, L., A generalization of Griffiths’ theorem on rational integrals II, Michigan Math. J. 58 (2009), 603–625.Google Scholar
Dimca, A. and Sticlaru, G., A computational approach to Milnor fiber cohomology, Forum Math. 29(4) (2017), 831–846.Google Scholar
Dimca, A. and Sticlaru, G., On the Milnor monodromy of the exceptional reflection arrangement of type $G_{31}$ , preprint, 2016, arXiv:1606.06615.Google Scholar
Durfee, A. H. and Hain, R. M., Mixed Hodge structures on the homotopy of links, Math. Ann. 280 (1988), 69–83.Google Scholar
Durfee, A. H. and M., Saito, Mixed Hodge structures on the intersection cohomology of links, Compos. Math. 76 (1990), 49–67.Google Scholar
Esnault, H., Fibre de Milnor d’un cône sur une courbe plane singulière, Invent. Math. 68 (1982), 477–496.Google Scholar
Hulek, K. and Kloosterman, R., Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces., Ann. Inst. Fourier (Grenoble) 61(3) (2011), 1133–1179.Google Scholar
Kloosterman, R., Cuspidal plane curves, syzygies and a bound on the MW-rank, J. Algebra 375 (2013), 216–234.Google Scholar
Lehrer, G. I. and Taylor, D. E., Unitary Reflection Groups, Australian Mathematical Society Lecture Series, Volume 20 (Cambridge University Press, Cambridge, 2009).Google Scholar
Libgober, A., Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49(4) (1982), 833–851.Google Scholar
Libgober, A., Development of the theory of Alexander invariants in algebraic geometry, in Topology of Algebraic Varieties and Singularities, Contemporary Mathematics, Volume 538, pp. 3–17 (American Mathematical Society, Providence, RI, 2011).Google Scholar
Măcinic, A. and Papadima, S., On the monodromy action on Milnor fibers of graphic arrangements, Topol. Appl. 156 (2009), 761–774.Google Scholar
Măcinic, A., Papadima, S. and Popescu, C. R., Modular equalities for complex reflexion arrangements, Doc. Math. 22 (2017), 135–150.Google Scholar
Mustaţă, M. and Popa, M., Hodge ideals, preprint, 2016, arXiv:1605.08088.Google Scholar
Navarro Aznar, V., Sur la théorie de Hodge-Deligne, Invent. math. 90 (1987), 11–76.Google Scholar
Oka, M., A survey on Alexander polynomials of plane curves, in Singularités Franco-Japonaises, Séminaires et Congrès., Volume 10, pp. 209–232 (Société Mathématique de France, Paris, 2005).Google Scholar
Orlik, P. and Terao, H., Arrangements of Hyperplanes (Springer, Berlin Heidelberg New York, 1992).Google Scholar
Papadima, S. and Suciu, A. I., The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy, Proc. Lond. Math. Soc. (3) 114(6) (2017), 961–1004.Google Scholar
Randell, R., Milnor fibers and Alexander polynomials of plane curves, in Singularities, Part 2 (Arcata, CA, 1981), pp. 415–419 (American Mathematical Society, Providence, RI, 1983).Google Scholar
Settepanella, S., A stability like theorem for cohomology of pure braid groups of the series A, B and D, Topol. Appl. 139(1) (2004), 37–47.Google Scholar
Settepanella, S., Cohomology of pure braid groups of exceptional cases, Topol. Appl. 156(5) (2009), 1008–1012.Google Scholar
Suciu, A., Fundamental groups of line arrangements: enumerative aspects, in Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), Contemporary Mathematics, Volume 276, pp. 43–79 (American Mathematical Society, Providence, RI, 2001).Google Scholar
Suciu, A., Hyperplane arrangements and Milnor fibrations, Ann. Fac. Sci. Toulouse Math. 23(2) (2014), 417–481.Google Scholar
Yoshinaga, M., Milnor fibers of real line arrangements, J. Singul. 7 (2013), 220–237.Google Scholar