Eigenvalue Of Monodromy Research Articles

On the monodromy conjecture for non-degenerate hypersurfaces

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein–Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations – the Denef– Loeser conjecture on topological zeta functions – is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of three variables is the existence of degenerate singularities arbitrarily close to a non-degenerate one. Thus, even aiming at the study of non-degenerate singularities, we have to go beyond this setting. We develop new tools to deal with such multidimensional phenomena, and conjecture how the proof for non-degenerate singularities of arbitrarily many variables might look like.

The monodromy conjecture for a space monomial curve with a plane semigroup

This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded Q-resolution of this pair and use an A’Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves.

Note on the monodromy conjecture for a space monomial curve with a plane semigroup

Roughly speaking, the monodromy conjecture for a singularity states that every pole of its motivic Igusa zeta function induces an eigenvalue of its monodromy. In this note, we determine both the motivic Igusa zeta function and the eigenvalues of monodromy for a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a plane branch. In particular, this yields a proof of the monodromy conjecture for such a curve.

On the universality of late-time correlators in semi-classical 2d CFTs

In the framework of the AdS3/ CFT2 correspondence, we present a systematic analysis of the late time thermalization of a two dimensional CFT state created by insertion of small number of heavy operators on the vacuum. We show that at late Lorentzian time, the universal features of this thermalization are solely captured by the eigenvalues of the monodromy matrix corresponding to the solutions of the uniformization equation. We discuss two different ways to extract the monodromy eigenvalues while bypassing the need for finding explicitly the full monodromy matrix - first, using a monodromy preserving diffeomorphism and second using Chen-Simons formulation of gravity in AdS3. Both of the methods yield the same precise relation between the eigenvalues and the final black hole temperature at late Lorentzian time.

THE HOLOMORPHY CONJECTURE FOR NONDEGENERATE SURFACE SINGULARITIES

The holomorphy conjecture roughly states that Igusa’s zeta function associated to a hypersurface and a character is holomorphic on$\mathbb{C}$whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over$\mathbb{C}$with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by$B_{1}$-facets.

A computational approach to Milnor fiber cohomology

Abstract In this paper we consider the Milnor fiber F associated to a reduced projective plane curve C. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of F, also known as the Alexander polynomial of the curve C, is presented. This leads to an effective algorithm to detect all the monodromy eigenvalues and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms.

Bernstein-Sato ideals and local systems

The topology of smooth quasi-projective complex varieties is very restrictive. One aspect of this statement is the fact that natural strata of local systems, called cohomology support loci, have a rigid structure: they consist of torsion-translated subtori in a complex torus. We propose and partially confirm a relation between Bernstein-Sato ideals and local systems. This relation gives yet a different point of view on the nature of the structure of cohomology support loci of local systems. The main result is a partial generalization to the case of a collection of polynomials of the theorem of Malgrange and Kashiwara which states that the Bernstein-Sato polynomial of a hypersurface recovers the monodromy eigenvalues of the Milnor fibers of the hypersurface. We also address a multi-variable version of the Monodromy Conjecture, prove that it follows from the usual single-variable Monodromy Conjecture, and prove it in the case of hyperplane arrangements.

Geometric criteria for tame ramification

We prove an A’Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field K. As a first application, we relate the orders of the tame monodromy eigenvalues on the l-adic cohomology of a K-curve to the geometry of a relatively minimal sncd-model, and we show that the semi-stable reduction theorem and Saito’s criterion for cohomological tameness are immediate consequences of this result. As a second application, we compute the error term in the trace formula for smooth and proper K-varieties. We see that the validity of the trace formula would imply a partial generalization of Saito’s criterion to arbitrary dimension.

Degenerating Riemann Surfaces and the Quillen Metric

The degeneration of the Quillen metric for a one-parameter family of Riemann surfaces has been studied by Bismut–Bost and Yoshikawa. In this article, we propose a more geometric point of view using Deligne’s Riemann–Roch theorem. We obtain an interpretation of the singular part of the metric as a discriminant and the continuous part as a degeneration of the metric on Deligne products, which gives an asymptotic development involving the monodromy eigenvalues. This generalizes the results of Bismut–Bost and is a version of Yoshikawa’s results on the degeneration of the Quillen metric for general degenerations with isolated singularities in the central fiber.

Generalized Monodromy Conjecture in dimension two

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.

Motivic zeta functions of abelian varieties, and the monodromy conjecture

We prove for abelian varieties a global form of Denef and Loeserʼs motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety A over a complete discretely valued field with algebraically closed residue field, its motivic zeta function has a unique pole at Chaiʼs base change conductor c ( A ) of A, and that the order of this pole equals one plus the potential toric rank of A. Moreover, we show that for every embedding of Q ℓ in C , the value exp ( 2 π i c ( A ) ) is an ℓ-adic tame monodromy eigenvalue of A. The main tool in the paper is Edixhovenʼs filtration on the special fiber of the Néron model of A, which measures the behavior of the Néron model under tame base change.

Jumps and monodromy of abelian varieties

We prove a strong form of the motivic monodromy conjecture for abelian varieties, by showing that the order of the unique pole of the motivic zeta function is equal to the maximal rank of a Jordan block of the corresponding monodromy eigenvalue. Moreover, we give a Hodge-theoretic interpretation of the fundamental invariants appearing in the proof.

Monodromy Jordan blocks, b-functions and poles of zeta functions for germs of plane curves

We study the poles of several local zeta functions: the Igusa, topological and motivic zeta function associated to a germ of a holomorphic function in two variables. It was known that there is at most one double pole for (any of) these zeta functions which is then given by the log canonical threshold of the function at the singular point. If the germ is reduced Loeser showed that such a double pole always induces a monodromy eigenvalue with a Jordan block of size 2. Here we settle the non-reduced situation, describing precisely in which case such a Jordan block of maximal size 2 occurs. We also provide detailed information about the Bernstein–Sato polynomial in the relevant non-reduced situation, confirming a conjecture of Igusa, Denef and Loeser.

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

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

On ‘maximal’ poles of zeta functions, roots ofb-functions, and monodromy Jordan blocks

The main objects of study in this paper are the poles of several local zeta functions: the Igusa, topological and motivic zeta function associated to a polynomial or (germ of) holomorphic function in n variables. We are interested in poles of maximal possible order n. In all known cases (curves, non-degenerate polynomials) there is at most one pole of maximal order n which is then given by the log canonical threshold of the function at the corresponding singular point. For an isolated singular point we prove that if the log canonical threshold yields a pole of order n of the corresponding (local) zeta function, then it induces a root of the Bernstein-Sato polynomial of the given function of multiplicity n (proving one of the cases of the strongest form of a conjecture of Igusa-Denef-Loeser). For an arbitrary singular point we show under the same assumption that the monodromy eigenvalue induced by the pole has a Jordan block of size n on the (perverse) complex of nearby cycles.

Zeta Functions and Monodromy for Surfaces that are General for a Toric Idealistic Cluster

In this article we consider surfaces that are general with respect to a 3- dimensional toric idealistic cluster. In particular, this means that blowing up a toric con- stellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part we prove the monodromy and the holomorphy conjecture for these surfaces.

Monodromy eigenvalues and zeta functions with differential forms

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 | 2 s 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 ( 2 π −1 s 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.

Sur la monodromie du problème de Cauchy ramifié

In this paper, we study the monodromy of the ramified Cauchy problem for operators with multiple characteristics of constant multiplicity. More precisely, we give an estimation of the eigenvalues of the solution's monodromy, first with the assumptions of the theorem of Hamada–Leray–Wagschal, then with the assumptions of the theorem of Leichtnam.

Monodromie du problème de Cauchy ramifié

In this Note, we study the monodromy of the ramified Cauchy problem for operators with multiple characteristics of constant multiplicity. More precisely, we give an estimation of the eigenvalues of the solution's monodromy, first with the assumptions of the theorem of Hamada–Leray–Wagschal, then with the assumptions of the theorem of Leichtnam. To cite this article: R. Camalès, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 639–642.

On the holomorphy conjecture for Igusa’s local zeta function

To a polynomial f over a p-adic field K and a character χ \chi of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z ( s , f , χ ) Z(s,f,\chi ) , which is a meromorphic function on C \mathbb {C} . Several theorems and conjectures relate the poles of Z ( s , f , χ ) Z(s,f,\chi ) to the monodromy of f; the so-called holomorphy conjecture states roughly that if the order of χ \chi does not divide the order of any eigenvalue of monodromy of f, then Z ( s , f , χ ) Z(s,f,\chi ) is holomorphic on C \mathbb {C} . We prove mainly that if the holomorphy conjecture is true for f ( x 1 , … , x n − 1 ) f({x_1}, \ldots ,{x_{n - 1}}) , then it is true for f ( x 1 , … , x n − 1 ) + x n k f({x_1}, \ldots ,{x_{n - 1}}) + x_n^k with k ≥ 3 k \geq 3 , and we give some applications.