Formulas for monodromy

TL;DR

This paper develops explicit formulas using tropical geometry to compute motivic nearby fibers and refined limit mixed Hodge numbers for degenerating complex varieties, with applications to monodromy actions.

Contribution

It introduces a stratification-based method and combinatorial formulas for monodromy invariants in degenerating families, extending previous results like Varchenko's formula.

Findings

Explicit formula for motivic nearby fiber via stratification and tropical geometry.

Combinatorial formula for refined limit mixed Hodge numbers for non-degenerate hypersurfaces.

Generalization of Varchenko's formula for monodromy Jordan block structure.

Abstract

Given a family $X$ of complex varieties degenerating over a punctured disc, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of $X$. Our first main result is that the motivic nearby fiber of $X$ can be computed by first stratifying $X$ into locally closed subvarieties that are non-degenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry. Our second main result is an explicit combinatorial formula for the refined limit mixed Hodge numbers in the case when $X$ is a family of non-degenerate hypersurfaces. As an application, given a complex polynomial, then, under appropriate conditions, we give a combinatorial formula for the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber, generalizing a famous formula of Varchenko for the associated eigenvalues. In addition, we give a formula for the Jordan block structure of the action of monodromy at infinity.