The holomorphy conjecture for nondegenerate surface singularities

TL;DR

This paper proves the holomorphy conjecture for nondegenerate surface singularities by relating monodromy eigenvalues to Newton polyhedron geometry and analyzing character sums to identify false poles.

Contribution

It establishes the holomorphy conjecture for a class of surface singularities and links monodromy eigenvalues to Newton polyhedron faces, including false pole analysis.

Findings

Holomorphy conjecture proven for nondegenerate surface singularities.

Relation between normalized volume of faces and monodromy eigenvalues.

Identification and elimination of false poles in zeta functions.

Abstract

The holomorphy conjecture states roughly that Igusa's zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article we prove the holomorphy conjecture for surface singularities which are nondegenerate over $\mathbb{C}$ with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volume (which appears in the formula of Varchenko for the zeta function of monodromy) of faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast with the context of the trivial character, we here need to show fakeness of certain poles in addition to the candidate poles contributed by $B_1$-facets.