Abstract
The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with a complex analytic function germ f defined on a normal surface singularity (X,0). The article targets the `right' extension in the case when the link of (X,0) is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function Z(f,w;s) for any f and analytic differential form w, which will play the key technical localization tool in the later definitions and proofs. Then, we define a set of `allowed' differential forms via a local restriction along each splice component. For plane curves we show the following three guiding properties: (1) if s' is any pole of Z(f,w;s) with w allowed, then exp(2\pi is') is a monodromy eigenvalue of f; (2) the `standard' form is allowed; (3) every monodromy eigenvalue of f is obtained as in (1) for some allowed w and some s'. For general (X,0) we prove (1) unconditionally, and (2)-(3) under an additional (necessary) assumption, which generalizes the semigroup condition of Neumann-Wahl. Several examples illustrate the definitions and support the basic assumptions.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.