The holomorphy conjecture for ideals in dimension two

TL;DR

This paper extends the holomorphy conjecture to arbitrary subschemes in two dimensions, proving it for ideals generated by finitely many complex polynomials in two variables, linking local monodromy and zeta functions.

Contribution

It formulates the holomorphy conjecture for topological zeta functions of arbitrary subschemes and proves it for two-dimensional ideals generated by finitely many polynomials.

Findings

Proposed the holomorphy conjecture for subschemes at the topological level.

Proved the conjecture for ideals in two variables generated by finitely many polynomials.

Established a connection between monodromy eigenvalues and the holomorphy of zeta functions.

Abstract

The holomorphy conjecture predicts that the local Igusa 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 note we propose the holomorphy conjecture for arbitrary subschemes at the level of the topological zeta function and we prove this conjecture for subschemes defined by an ideal that is generated by a finite number of complex polynomials in two variables.