The local motivic monodromy conjecture for simplicial nondegenerate singularities

TL;DR

This paper proves the local motivic monodromy conjecture for a class of singularities characterized by simplicial Newton polyhedra, linking poles of zeta functions to monodromy eigenvalues.

Contribution

It establishes the conjecture for nondegenerate simplicial singularities, connecting motivic zeta functions with monodromy eigenvalues.

Findings

Poles of local topological zeta functions correspond to monodromy eigenvalues.

The result applies to Igusa's local p-adic zeta functions for large primes p.

The proof confirms the conjecture for a broad class of singularities.

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 $p$.