[Paper Review] The monodromy conjecture for zeta functions associated to ideals in dimension two

Lise Van Proeyen, Willem Veys|ArXiv.org|Oct 12, 2009

Algebraic Geometry and Number Theory24 references3 citations

TL;DR

This paper proves a generalized monodromy conjecture for topological zeta functions associated to ideals in two complex variables, showing that every pole corresponds to an eigenvalue of the Verdier monodromy action. Using embedded principalization and numerical data from resolution of singularities, the authors establish that poles at $-\frac{\nu_i}{N_i}$ correspond to monodromy eigenvalues $e^{2\pi i s_0}$, extending the classical conjecture from polynomials to ideals.

ABSTRACT

The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta functions associated to an ideal. First we work in arbitrary dimension and obtain a formula (like the one of A'Campo) to compute the 'Verdier monodromy' eigenvalues associated to an ideal. Afterwards we prove a generalized monodromy conjecture for arbitrary ideals in two variables.

Motivation & Objective

• To extend the monodromy conjecture from zeta functions of single polynomials to those associated with ideals in dimension two.
• To define and compute 'Verdier monodromy' eigenvalues for ideals in two variables, generalizing the classical Milnor monodromy.
• To establish a correspondence between poles of the topological zeta function and eigenvalues of monodromy action under the generalized framework.
• To verify the conjecture using explicit resolution data and numerical invariants from principalizations.

Proposed method

• Construct the topological zeta function for an ideal $\mathcal{I} = (f_1, \dots, f_r)$ in $\mathbb{C}[x,y]$ via embedded principalization of the ideal.
• Use the resolution data—specifically the numerical invariants $\nu_i$ and $N_i$—to compute the poles of the zeta function as $-\frac{\nu_i}{N_i}$.
• Define the Verdier monodromy action on the cohomology of the fiber over points of the exceptional divisor in the resolution.
• Analyze the monodromy zeta functions at generic points of the exceptional divisor to extract eigenvalues $e^{2\pi i s_0}$.
• Apply the Weak Factorization Theorem to ensure independence of the zeta function on the choice of resolution.
• Use contradiction arguments on infinite chains of exceptional curves to prove that only finitely many components can contribute to poles.

Experimental results

Research questions

• RQ1 Does every pole of the topological zeta function associated to an ideal in two variables correspond to an eigenvalue of the Verdier monodromy action?
• RQ2 Can the classical monodromy conjecture be extended from polynomials to ideals in dimension two?
• RQ3 How do the numerical invariants $\nu_i$ and $N_i$ from a principalization relate to monodromy eigenvalues for ideals?
• RQ4 What is the structure of the monodromy zeta function at generic points of the exceptional divisor in the resolution of an ideal?
• RQ5 Are the poles of the zeta function stable under different choices of resolution, and how does this affect the monodromy correspondence?

Key findings

• The monodromy conjecture holds for all ideals in two complex variables: every pole $s_0 = -\frac{\nu_i}{N_i}$ of the topological zeta function corresponds to an eigenvalue $e^{2\pi i s_0}$ of the Verdier monodromy action.
• For the ideal $(x^3y, x^6 + y^4)$, the poles $-\frac{1}{2}$ and $-\frac{2}{3}$ correspond to monodromy eigenvalues $e^{-\pi i}$ and $e^{-\frac{4\pi i}{3}}$, respectively.
• For the ideal $(x^4, xy^2, y^3)$, the poles $-\frac{2}{3}$ and $-\frac{5}{8}$ yield monodromy eigenvalues $e^{-\frac{4\pi i}{3}}$ and $e^{-\frac{5\pi i}{4}}$, confirming the conjecture.
• In the case of $(x^3y, x^3 - y^2)$, the poles $-\frac{5}{6}$ and $-\frac{8}{9}$ produce eigenvalues $e^{-\frac{5\pi i}{3}}$ and $e^{-\frac{16\pi i}{9}}$, respectively.
• The proof relies on contradiction arguments involving infinite chains of intersecting exceptional curves, showing that only finitely many components can contribute to poles.
• The result extends to motivic, Hodge, and $p$-adic Igusa zeta functions due to shared pole conditions under the same resolution framework.