Abstract
A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type \(G_{31}\).
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Bailet, P., Yoshinaga, M.: Degeneration of Orlik–Solomon algebras and Milnor fibers of complex line arrangements. Geom. Dedicata 175, 49–56 (2015)
Choudary, A.D.R., Dimca, A., Papadima, S.: Some analogs of Zariski theorem on nodal line arrangements. Algebr. Geom. Topol. 5, 691–711 (2005)
Cohen, D., Dimca, A., Orlik, P.: Nonresonance conditions for arrangements. Ann. Institut Fourier (Grenoble) 53, 1883–1896 (2003)
Dimca, A.: Singularities and topology of hypersurfaces. Universitext. Springer, New York (1992)
Dimca, A.: Sheaves in Topology. Universitext. Springer, Berlin (2004)
Dimca, A.: Hyperplane Arrangements: An Introduction. Universitext. Springer, Cham (2017)
Dimca, A.: On the Milnor monodromy of the irreducible complex reflection arrangements. J. Inst. Math. Jussieu (2017). https://doi.org/10.1017/S147474801700038X
Dimca, A., Sticlaru, G.: On the Milnor monodromy of the exceptional reflection arrangement of type \(G_{31}\), arXiv: 1606.06615
Goodman, J.E.: Affine open subsets of algebraic varieties and ample divisors. Ann. Math. 89, 160–183 (1969)
Hartshorne, R.: Cohomological dimension of algebraic varieties. Ann. Math. 88, 403–450 (1968)
Hartshorne, R.: Algebraic Geometry. GTM, vol. 52. Springer, Heidelberg (1977)
Hoge, T., Röhrle, G.: On supersolvable reflection arrangements. Proc. AMS 142(11), 3787–3799 (2014)
Libgober, A.: Eigenvalues for the monodromy of the Milnor fibers of arrangements. In: Libgober, A., Tibăr, M. (eds.) Trends in Mathematics: Trends in Singularities. Birkhäuser, Basel (2002)
Matsuki, K.: Introduction to the Mori Program. Universitext. Springer, New York (2002)
Măcinic, A., Papadima, S.: On the monodromy action on Milnor fibers of graphic arrangements. Topology and its Applications 156, 761–774 (2009)
Măcinic, A., Papadima, S., Popescu, C.R.: Modular equalities for complex reflexion arrangements. Documenta Math. 22, 135–150 (2017)
Nazir, S., Yoshinaga, M.: On the connectivity of the realization spaces of line arrangements. Annali della Scuola Normale Superiore di Pisa 11, 921–937 (2012)
Orlik, P., Terao, H.: Arrangements of Hyperplanes. Springer, Berlin, Heidelberg, New York (1992)
Randell, R.: Lattice-isotopic arrangements are topologically isomorphic. Proc. Am. Math. Soc. 107(2), 555–559 (1989)
Salvetti, M., Serventi, M.: On the twisted cohomology of affine line arrangements. Configuration spaces, Springer INdAM Ser. 14: 275–290 (2016)
Salvetti, M., Serventi, M.: Arrangements of lines and monodromy of associated Milnor fibers. J. Knot Theory Ramifications. 25(12), 1642014 (2016)
Yoshinaga, M.: Milnor fibers of real line arrangements. J. Singul. 7, 220–237 (2013)
Yoshinaga, M.: Resonant bands and local system cohomology groups for real line arrangements. Vietnam J. Math. 42(3), 377–392 (2014)
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Bailet, P., Dimca, A. & Yoshinaga, M. A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements. manuscripta math. 157, 497–511 (2018). https://doi.org/10.1007/s00229-018-0999-y
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DOI: https://doi.org/10.1007/s00229-018-0999-y