Monodromy conjecture for some surface singularities (original) (raw)

Monodromy eigenvalues are induced by poles of zeta functions: the irreducible curve case

Bulletin of the London Mathematical Society, 2010

The 'monodromy conjecture' for a hypersurface singularity f predicts that a pole of its topological (or related) zeta function induces one of its monodromy eigenvalues. However, in general only a few eigenvalues are obtained this way. The second author proposed to consider zeta functions associated with the hypersurface and with a differential form and raised the following question. Can one find a list of differential forms ω i such that any pole of the zeta function of f and an ω i induces a monodromy eigenvalue of f , and such that all monodromy eigenvalues of f are obtained this way? Here we provide an affirmative answer for an arbitrary irreducible curve singularity f .

On the poles of topological zeta functions

Proceedings of the American Mathematical Society

We study the topological zeta function Z_{top,f}(s) associated to a polynomial f with complex coefficients. This is a rational function in one variable and we want to determine the numbers that can occur as a pole of some topological zeta function; by definition these poles are negative rational numbers. We deal with this question in any dimension. Denote P_n := {s_0 | \exists f in C[x_1,..., x_n] : Z_{top,f}(s) has a pole in s_0}. We show that {-(n-1)/2-1/i | i in Z_{>1}} is a subset of P_n; for n=2 and n=3, the last two authors proved before that these are exactly the poles less then -(n-1)/2. As main result we prove that each rational number in the interval [-(n-1)/2,0) is contained in P_n.

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.

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.

On atypical values and local monodromies of meromorphic functions

Arxiv preprint math/ …, 1998

A meromorphic function on a compact complex analytic manifold defines a C ∞ locally trivial fibration over the complement of a finite set in the projective line CP 1. We describe zeta-functions of local monodromies of this fibration around atypical values. Some applications to polynomial functions on C n are described. §1.-Introduction We want to consider fibrations defined by meromorphic functions. In order to have more general statements we prefer to use the notion of a meromorphic function slightly different from the standard one. Let M be an n-dimensional compact complex analytic manifold.

Monodromy eigenvalues and zeta functions with differential forms

Advances in Mathematics, 2007

For a complex polynomial or analytic function f , there is a strong correspondence between poles of the so-called local zeta functions or complex powers |f | 2s ω, where the ω are C ∞ differential forms with compact support, and eigenvalues of the local monodromy of f . In particular Barlet showed that each monodromy eigenvalue of f is of the form exp(2π √ −1s 0 ), where s 0 is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.

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.