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Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

Published online by Cambridge University Press:  11 January 2016

Alexandru Dimca
Alexandru Dimca*
Affiliation:
Institut Universitaire de France et Laboratoire J.A. Dieudonné, UMR CNRS 7351 Université de Nice Sophia Antipolis 06108 Nice Cedex 02, France, [email protected]
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Abstract

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The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.

It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.

We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Keywords

32S2232S3532S2532S55
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 206 , June 2012 , pp. 75 - 97
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

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