On a Generalized Monodromy Conjecture for Curves using Differential Forms

TL;DR

This paper investigates generalized versions of the monodromy conjecture for complex surface germs and affine planes, exploring the relationship between zeta function poles and monodromy eigenvalues, and providing counterexamples.

Contribution

It introduces generalized monodromy conjectures for singular surfaces and intrinsic forms, and reveals a link between poles and polar curve intersections in specific cases.

Findings

Counterexamples to generalized conjectures are provided.

A relation between zeta function poles and polar curve intersections is established.

The study extends understanding of monodromy conjecture beyond standard forms.

Abstract

Motivic and topological zeta functions are singularity invariants, mainly associated to a function $f$ and a top differential form $\omega$ on a smooth variety. When $\omega$ is the standard form $dx_1\wedge \dots \wedge dx_n$ on affine $n$-space, the monodromy conjecture states that poles of these zeta functions should induce monodromy eigenvalues of $f$. We study natural generalized statements of the monodromy conjecture for functions $f$ on complex surface germs; more precisely on singular surfaces for forms $\omega$ that generalize the standard form, and on the affine plane for forms $\omega$ that are intrinsically associated to $f$. For all cases, we provide counterexamples to the statement. In addition, when the intrinsically associated $\omega$ is given by the generic polar of $f$, we discover a relation between the poles of the zeta functions and the intersection behaviour of the polar curve.