The ring of local tropical fans and tropical nearby monodromy eigenvalues

TL;DR

This paper extends tropical intersection theory to germs of analytic sets, introduces tropical characteristic classes, and explores their connection to monodromy eigenvalues and zeta functions, proposing a stronger conjecture linking poles to tropical eigenvalues.

Contribution

It develops a local version of the ring of tropical fans, introduces tropical characteristic classes, and formulates a stronger monodromy conjecture relating poles of zeta functions to tropical eigenvalues.

Findings

Expressed some monodromy eigenvalues in terms of resolutions and exceptional divisors.

Proposed a stronger conjecture linking poles of zeta functions to tropical eigenvalues.

Confirmed the conjecture for non-degenerate singularities in up to 4 variables.

Abstract

We extend the tropical intersection theory to tropicalizations of germs of analytic sets. In particular, we construct a (not entirely obvious) local version of the ring of tropical fans with a nondegenerate intersection pairing. As an application, we study nearby monodromy eigenvalues -- the eigenvalues of the monodromy operators of singularities, adjacent to a given singularity of a holomorphic function $f$. More precisely, we express some of such values in terms of certain resolutions of $f$. The expression is given it terms of the exceptional divisor strata of arbitrary codimension, generalizing the classical A'Campo formula that consumes only codimension 1 strata and produces only monodromy eigenvalues at the origin. For this purpose, we introduce tropical characteristic classes of germs of analytic sets, and use this calculus to detect some of the nearby monodromy eigenvalues, which we call tropical. The study is motivated by the monodromy conjectures by Igusa, Denef and Loeser: every pole of an appropriate local zeta function of $f$ induces a nearby monodromy eigenvalue. We propose a presumably stronger version of this conjecture: all poles of the local zeta function induce tropical nearby monodromy eigenvalues. In particular, if the singularity is non-degenerate with respect to its Newton polyhedron $N$, then the tropical monodromy eigenvalues can be expressed in terms of fiber polytopes of certain faces of $N$, so our conjecture (unlike the original ones) becomes a purely combinatorial statement about a polyhedron. This statement is confirmed for the topological zeta function in dimension up to 4 in a joint work with A. Lemahieu and K. Takeuchi, which, in particular, supports our conjecture and proves the original one for non-degenerate singularities in 4 variables.