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 $\mathbb{Q}$, 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 $\mathbb{Q}$, 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.