Zeta Functions for Curves and Log Canonical Models (original) (raw)

Topological zeta functions of complex plane curve singularities

2020

We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.

On the Poles of Maximal Order of the Topological Zeta Function

Bulletin of the London Mathematical Society, 1999

The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By de nition these zeta functions are rational functions in one variable and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non degenerate with respect to its Newton polyhedron we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; concerning the global zeta function we give a similar result. Moreover for any f we show that poles of maximal possible order are always of the form ?1=N with N a positive integer.

Topological zeta functions and the monodromy conjecture for complex plane curves

arXiv (Cornell University), 2020

We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.

Zeta Functions in Algebra and Geometry

Contemporary Mathematics, 2012

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Deformations of polynomials and their zeta-functions

Journal of Mathematical Sciences, 2007

For an analytic in σ ∈ (C, 0) family P σ of polynomials in n variables there is defined a monodromy transformation h of the zero level set V σ = {P σ = 0} for σ = 0 small enough. The zeta function of this monodromy transformation is written as an integral with respect to the Euler characteristic of the corresponding local data. This leads to a study of deformations of holomorphic germs and their zeta functions. We show some examples of computations with the use of this technique. *

THE TOPOLOGICAL ZETA FUNCTION ASSOCIATED TO A FUNCTION ON A NORMAL SURFACE GERM

Topology, 1999

We associate to a regular function f on a normal surface germ (S; 0) an invariant, called the topological zeta function, which generalizes the same invariant for a plane curve germ; by de nition it is a rational function in one variable. We study its poles and their relation with the local monodromy of f , in particular we prove the`generalized holomorphy conjecture'. We give a formula for this topological zeta function in terms of the log canonical model of (S; f ?1 f0g), and we also introduce a still more general invariant.