Geometry & Topology

Generalized Monodromy Conjecture in dimension two

András Némethi and Willem Veys

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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.

Article information

Source
Geom. Topol., Volume 16, Number 1 (2012), 155-217.

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732382

Digital Object Identifier
doi:10.2140/gt.2012.16.155

Mathematical Reviews number (MathSciNet)
MR2872581

Zentralblatt MATH identifier
1241.14004

Subjects
Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14H20: Singularities, local rings [See also 13Hxx, 14B05] 32S40: Monodromy; relations with differential equations and D-modules
Secondary: 32S05: Local singularities [See also 14J17] 14H50: Plane and space curves 14J17: Singularities [See also 14B05, 14E15] 32S25: Surface and hypersurface singularities [See also 14J17]

Keywords
monodromy conjecture topological zeta function monodromy surface singularity plane curve singularity resolution graph semigroup condition splice diagram splice decomposition

Citation

Némethi, András; Veys, Willem. Generalized Monodromy Conjecture in dimension two. Geom. Topol. 16 (2012), no. 1, 155--217. doi:10.2140/gt.2012.16.155. https://projecteuclid.org/euclid.gt/1513732382


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