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Cohomology of the Milnor Fibre of a Hyperplane Arrangement with Symmetry

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Configuration Spaces

Part of the book series: Springer INdAM Series ((SINDAMS,volume 14))

Abstract

We prove some general results concerning the cohomology of the Milnor fibre of a hyperplane arrangement, and apply them to the case when the arrangement has some symmetry properties, particularly the case of the set of reflecting hyperplanes of a unitary reflection group. We relate the isotypic components of the monodromy action on the cohomology to the cohomology degree and to the mixed Hodge structure of the cohomology. We also use monodromy eigenspaces to determine the spectrum in some cases, which in turn throws further light on the equivariant Hodge structure of the cohomology and on the determination of the equivariant Hodge-Deligne polynomials. When the arrangement is the set of reflecting hyperplanes of a unitary reflection group, then using eigenspace theory for reflection groups, we prove sum formulae for additive functions such as the equivariant weight polynomial and certain polynomials related to the Euler characteristic, such as the Hodge-Deligne polynomials. This leads to a case-free determination of the Euler characteristic in this case, answering a question of Denham-Lemire. We also give an alternative formula for the spectrum of an arrangement which permits its computation in low dimensions, and we provide several examples of such computations.

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Acknowledgements

The author “Alexandru Dimca” was partially supported by Institut Universitaire de France. The authors “Alexandru Dimca and Gus Lehrer” were partially supported by Australian Research Council Grants DP0559325 and DP110103451.

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Authors and Affiliations

  1. University of Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100, Nice, France

    Alexandru Dimca

  2. School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW, 2006, Australia

    Gus Lehrer

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  1. Alexandru Dimca
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  2. Gus Lehrer
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Correspondence to Gus Lehrer .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Filippo Callegaro

  2. Department of Mathematics, University of Rochester, Rochester, USA

    Frederick Cohen

  3. Dipartimento di Matematica, Università di Roma La Sapienza, Roma, Italy

    Corrado De Concini

  4. Department of Mathematics, University of Bremen, Bremen, Germany

    Eva Maria Feichtner

  5. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Giovanni Gaiffi

  6. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Mario Salvetti

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Dimca, A., Lehrer, G. (2016). Cohomology of the Milnor Fibre of a Hyperplane Arrangement with Symmetry. In: Callegaro, F., Cohen, F., De Concini, C., Feichtner, E., Gaiffi, G., Salvetti, M. (eds) Configuration Spaces. Springer INdAM Series, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-31580-5_10

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