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The local motivic monodromy conjecture for simplicial nondegenerate singularities

Matt Larson, Sam Payne, Alan Stapledon

TL;DR

This work proves the local motivic monodromy conjecture for nondegenerate singularities with simplicial Newton polyhedra by establishing a nonnegative combinatorial formula for nearby monodromy eigenvalues via local $h$-polynomials and Ehrhart data, and by introducing a local formal zeta function to systematically remove fake poles. Consequently, every candidate pole associated to a facet of the Newton polyhedron yields a nearby monodromy eigenvalue, yielding the local topological and (for large primes) $p$-adic monodromy conjectures in this setting. The method also extends to certain non-simplicial cases, includes a mechanism to handle poles through a formal zeta framework, and applies to dimension three to confirm the conjecture in that case. The results reveal the central role of simpliciality, $U B_1$-face structure, and local $h$-polynomials in governing the interplay between zeta functions and monodromy, while also identifying obstructions and counterexamples in higher dimensions that guide future work.

Abstract

We prove the local motivic monodromy conjecture for singularities that are nondegenerate with respect to a simplicial Newton polyhedron. It follows that all poles of the local topological zeta functions of such singularities correspond to eigenvalues of monodromy acting on the cohomology of the Milnor fiber of some nearby point, as do the poles of Igusa's local $p$-adic zeta functions for large primes $p$.

The local motivic monodromy conjecture for simplicial nondegenerate singularities

TL;DR

This work proves the local motivic monodromy conjecture for nondegenerate singularities with simplicial Newton polyhedra by establishing a nonnegative combinatorial formula for nearby monodromy eigenvalues via local -polynomials and Ehrhart data, and by introducing a local formal zeta function to systematically remove fake poles. Consequently, every candidate pole associated to a facet of the Newton polyhedron yields a nearby monodromy eigenvalue, yielding the local topological and (for large primes) -adic monodromy conjectures in this setting. The method also extends to certain non-simplicial cases, includes a mechanism to handle poles through a formal zeta framework, and applies to dimension three to confirm the conjecture in that case. The results reveal the central role of simpliciality, -face structure, and local -polynomials in governing the interplay between zeta functions and monodromy, while also identifying obstructions and counterexamples in higher dimensions that guide future work.

Abstract

We prove the local motivic monodromy conjecture for singularities that are nondegenerate with respect to a simplicial Newton polyhedron. It follows that all poles of the local topological zeta functions of such singularities correspond to eigenvalues of monodromy acting on the cohomology of the Milnor fiber of some nearby point, as do the poles of Igusa's local -adic zeta functions for large primes .
Paper Structure (48 sections, 71 theorems, 151 equations, 1 figure)

This paper contains 48 sections, 71 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Background and motivation
  4. Archimedean zeta functions
  5. Log resolutions and zeta functions of monodromy
  6. Igusa's local $\mathfrak{p}$-adic zeta functions
  7. Good reduction
  8. Local topological zeta functions
  9. The local motivic zeta function
  10. Prior results
  11. Local monodromy conjectures
  12. Global zeta functions and strong monodromy conjectures
  13. Further variants of the local zeta functions and monodromy conjectures
  14. Methods and structure of the paper
  15. Key definitions
  16. ...and 33 more sections

Key Result

Theorem 1.1.1

Suppose that $\mathop{\mathrm{Newt}}\nolimits(f)$ is simplicial and $f$ is nondegenerate. Then there is a set of candidate poles $\mathcal{P} \subset \mathbb{Q}$ for $Z_\mathrm{mot}(T)$ such that, for every $\alpha \in \mathcal{P}$, $\exp(2\pi i \alpha)$ is a nearby eigenvalue of monodromy.

Figures (1)

  • Figure 1: A complete fan with maximal cones indexed by faces containing $M$ and a corresponding deformation. We show the intersection of $\mathop{\mathrm{span}}\nolimits(\sigma_M)$ with an affine hyperplane. The cone $\sigma_M$ is shown in black and grey, while the codimension $1$ cones of the complete fan appear in red, with their maximal cones labeled in red.

Theorems & Definitions (194)

  • Theorem 1.1.1
  • Remark 1.1.2
  • Theorem 1.1.3
  • Theorem 1.1.4
  • Theorem 1.2.1: Malgrange74
  • Theorem 1.2.2: ACampo75
  • Remark 1.2.3
  • Definition 1.2.4
  • Theorem 1.2.5: Denef87
  • Remark 1.2.6
  • ...and 184 more