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ON THE MILNOR MONODROMY OF THE IRREDUCIBLE COMPLEX REFLECTION ARRANGEMENTS

  • Alexandru Dimca (a1)
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 $G$ .

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1. 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.
2. Bailet P. and Settepanella S., Homology graph of real arrangements and monodromy of Milnor fiber, Adv. Appl. Math. 90 (2017), 4685.
3. Beltrametti M. and Sommese A.J., On k-jet ampleness, in Complex Analysis and Geometry (ed. Silva Ancona), pp. 355376 (Plenum Press, NY, 1993).
4. Bessis D., Finite complex reflection arrangements are K (𝜋, 1), Ann. of Math. (2) 181(2015) 809904.
5. Budur N., Dimca A. and Saito M., First Milnor cohomology of hyperplane arrangements, Contemp. Math. 538 (2011), 279292.
6. Cohen D. C. and Suciu A. I., On Milnor fibrations of arrangements, J. Lond. Math. Soc. 51(2) (1995), 105119.
7. 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), 645656.
8. Dimca A., Betti numbers of hypersurfaces and defects of linear systems, Duke Math. J. 60 (1990), 285298.
9. Dimca A., Singularities and Topology of Hypersurfaces, Universitext (Springer, New York, 1992).
10. Dimca A., Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements, Nagoya Math. J. 206 (2012), 7597.
11. Dimca A., Hyperplane Arrangements: An Introduction, Universitext (Springer, Cham, Switzerland, 2017).
12. 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. 231253 (Edizioni della Normale, Pisa, Italy, 2012).
13. 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).
14. Dimca A., Saito M. and Wotzlaw L., A generalization of Griffiths’ theorem on rational integrals II, Michigan Math. J. 58 (2009), 603625.
15. Dimca A. and Sticlaru G., A computational approach to Milnor fiber cohomology, Forum Math. 29(4) (2017), 831846.
16. Dimca A. and Sticlaru G., On the Milnor monodromy of the exceptional reflection arrangement of type inline-graphic $G_{31}$ , preprint, 2016, arXiv:1606.06615.
17. Durfee A. H. and Hain R. M., Mixed Hodge structures on the homotopy of links, Math. Ann. 280 (1988), 6983.
18. Durfee A. H. and M. Saito, Mixed Hodge structures on the intersection cohomology of links, Compos. Math. 76 (1990), 4967.
19. Esnault H., Fibre de Milnor d’un cône sur une courbe plane singulière, Invent. Math. 68 (1982), 477496.
20. 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), 11331179.
21. Kloosterman R., Cuspidal plane curves, syzygies and a bound on the MW-rank, J. Algebra 375 (2013), 216234.
22. Lehrer G. I. and Taylor D. E., Unitary Reflection Groups, Australian Mathematical Society Lecture Series, Volume 20 (Cambridge University Press, Cambridge, 2009).
23. Libgober A., Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49(4) (1982), 833851.
24. Libgober A., Development of the theory of Alexander invariants in algebraic geometry, in Topology of Algebraic Varieties and Singularities, Contemporary Mathematics, Volume 538, pp. 317 (American Mathematical Society, Providence, RI, 2011).
25. Măcinic A. and Papadima S., On the monodromy action on Milnor fibers of graphic arrangements, Topol. Appl. 156 (2009), 761774.
26. Măcinic A., Papadima S. and Popescu C. R., Modular equalities for complex reflexion arrangements, Doc. Math. 22 (2017), 135150.
27. Mustaţă M. and Popa M., Hodge ideals, preprint, 2016, arXiv:1605.08088.
28. Navarro Aznar V., Sur la théorie de Hodge-Deligne, Invent. math. 90 (1987), 1176.
29. Oka M., A survey on Alexander polynomials of plane curves, in Singularités Franco-Japonaises, Séminaires et Congrès., Volume 10, pp. 209232 (Société Mathématique de France, Paris, 2005).
30. Orlik P. and Terao H., Arrangements of Hyperplanes (Springer, Berlin Heidelberg New York, 1992).
31. 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), 9611004.
32. Randell R., Milnor fibers and Alexander polynomials of plane curves, in Singularities, Part 2 (Arcata, CA, 1981), pp. 415419 (American Mathematical Society, Providence, RI, 1983).
33. Settepanella S., A stability like theorem for cohomology of pure braid groups of the series A, B and D, Topol. Appl. 139(1) (2004), 3747.
34. Settepanella S., Cohomology of pure braid groups of exceptional cases, Topol. Appl. 156(5) (2009), 10081012.
35. 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. 4379 (American Mathematical Society, Providence, RI, 2001).
36. Suciu A., Hyperplane arrangements and Milnor fibrations, Ann. Fac. Sci. Toulouse Math. 23(2) (2014), 417481.
37. Yoshinaga M., Milnor fibers of real line arrangements, J. Singul. 7 (2013), 220237.
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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