Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements
Abstract
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 , 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 , whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has not polynomial count. Such examples are shown not to exist in low dimensions.
Cite
@article{arxiv.1012.1437,
title = {Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements},
author = {Alexandru Dimca},
journal= {arXiv preprint arXiv:1012.1437},
year = {2011}
}
Comments
In this new version some references are added for Thom-Sebastiani type results for the productof two functions. Note that all the previous results make no claim on the corresponding mixed Hodge structures, which is a key point in our paper