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2012 Generalized Monodromy Conjecture in dimension two
András Némethi, Willem Veys
Geom. Topol. 16(1): 155-217 (2012). DOI: 10.2140/gt.2012.16.155

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 an analytic germ f:(X,0)(,0) 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,ω;s) for any f and analytic differential form ω, 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 s0 is any pole of Z(f,ω;s) with ω allowed, then exp(2πis0) 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 ω and some s0.

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.

Citation

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András Némethi. Willem Veys. "Generalized Monodromy Conjecture in dimension two." Geom. Topol. 16 (1) 155 - 217, 2012. https://doi.org/10.2140/gt.2012.16.155

Information

Received: 1 February 2011; Accepted: 26 August 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1241.14004
MathSciNet: MR2872581
Digital Object Identifier: 10.2140/gt.2012.16.155

Subjects:
Primary: 14B05, 14H20, 32S40
Secondary: 14H50, 14J17, 32S05, 32S25

Rights: Copyright © 2012 Mathematical Sciences Publishers

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