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

TL;DR

This paper investigates the monodromy of Milnor fibers of hyperplane arrangements, establishing combinatorial criteria for triviality, and explores properties like Tate cohomology and polynomial count in relation to monodromy behavior.

Contribution

It provides a combinatorial criterion for the order of monodromy in hyperplane arrangements and constructs examples with specific properties relating to Tate cohomology and polynomial count.

Findings

Monodromy order is combinatorially determined.

Trivial monodromy implies Tate cohomology and polynomial count.

Existence of arrangements with nontrivial monodromy, Tate property, and no polynomial count.

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.