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On a Generalized Monodromy Conjecture for Curves using Differential Forms

Lise Fonteyne, Willem Veys

TL;DR

The paper tests generalized versions of the monodromy conjecture for curves on singular ambient spaces and for differential 2-forms intrinsically linked to the defining function, using embedded resolutions and local zeta data. Across normal and non-normal surface settings and for Hessian- and polar-based forms, it constructs explicit counterexamples showing that poles of the associated zeta functions do not always impose monodromy eigenvalues, thereby defeating broad generalizations of the classical conjecture. A notable finding is a link between poles from a generic polar and the intersection pattern of the polar curve with the exceptional locus, which suggests a refined geometric structure behind zeta-pole phenomena. These results delineate the limitations of extending the monodromy conjecture beyond the smooth plane context and motivate further investigation into polar geometry and intrinsic forms.

Abstract

Motivic and topological zeta functions are singularity invariants, mainly associated to a function $f$ and a top differential form $ω$ on a smooth variety. When $ω$ 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 $ω$ that generalize the standard form, and on the affine plane for forms $ω$ that are intrinsically associated to $f$. For all cases, we provide counterexamples to the statement. In addition, when the intrinsically associated $ω$ 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.

On a Generalized Monodromy Conjecture for Curves using Differential Forms

TL;DR

The paper tests generalized versions of the monodromy conjecture for curves on singular ambient spaces and for differential 2-forms intrinsically linked to the defining function, using embedded resolutions and local zeta data. Across normal and non-normal surface settings and for Hessian- and polar-based forms, it constructs explicit counterexamples showing that poles of the associated zeta functions do not always impose monodromy eigenvalues, thereby defeating broad generalizations of the classical conjecture. A notable finding is a link between poles from a generic polar and the intersection pattern of the polar curve with the exceptional locus, which suggests a refined geometric structure behind zeta-pole phenomena. These results delineate the limitations of extending the monodromy conjecture beyond the smooth plane context and motivate further investigation into polar geometry and intrinsic forms.

Abstract

Motivic and topological zeta functions are singularity invariants, mainly associated to a function and a top differential form on a smooth variety. When is the standard form on affine -space, the monodromy conjecture states that poles of these zeta functions should induce monodromy eigenvalues of . We study natural generalized statements of the monodromy conjecture for functions on complex surface germs; more precisely on singular surfaces for forms that generalize the standard form, and on the affine plane for forms that are intrinsically associated to . For all cases, we provide counterexamples to the statement. In addition, when the intrinsically associated is given by the generic polar of , we discover a relation between the poles of the zeta functions and the intersection behaviour of the polar curve.
Paper Structure (11 sections, 11 theorems, 38 equations, 9 figures)

This paper contains 11 sections, 11 theorems, 38 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Zeta functions
  4. Monodromy conjecture
  5. Setting
  6. Curves on singular surfaces
  7. Curves on normal surfaces
  8. Curves on a non-normal surface
  9. Plane curves
  10. Hessian
  11. Polar curves

Key Result

Theorem 1.1

Let $f$ be a non-constant regular function on a normal surface germ $(S,o)$, where $f(o)=0$. Suppose all the log discrepancies $\nu_j$, $j\in T_e,$ are integral. If $s_0 \in \mathbb{Q}_{\leq 0}$ is a pole of $Z_{\operatorname{top},o}(f;s)$ or $Z_{\operatorname{mot},o}(f;s)$, then $e^{2\pi i s_0}$ is

Figures (9)

  • Figure 1: Process of an embedded resolution of $x^4-y$
  • Figure 2: Intersection diagram for $y^{10}-z^{13}$
  • Figure 3: Intersection diagrams
  • Figure 4: Intersection diagram for $x^3-y^2$
  • Figure 5: Comparison of intersection diagrams
  • ...and 4 more figures

Theorems & Definitions (31)

  • Theorem 1.1: rodrigues
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Conjecture 2.6: Monodromy conjecture
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 21 more