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 $\Q$, 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 not polynomial count. Such examples are shown not to exist in low dimensions.

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