Monodromy Zeta Function Research Articles

Mirror symmetry on levels of non-abelian Landau–Ginzburg orbifolds

We consider the Berglund–Hübsch–Henningson–Takahashi duality of Landau–Ginzburg orbifolds with a symmetry group generated by some diagonal symmetries and some permutations of variables. We study the orbifold Euler characteristics, the orbifold monodromy zeta functions and the orbifold E-functions of such dual pairs. We conjecture that we get a mirror symmetry between these invariants even on each level, where we call level the conjugacy class of a permutation. We support this conjecture by giving partial results for each of these invariants.

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.

ON COMPLEX HOMOGENEOUS SINGULARITIES

We study the singularity at the origin of $\mathbb{C}^{n+1}$ of an arbitrary homogeneous polynomial in $n+1$ variables with complex coefficients, by investigating the monodromy characteristic polynomials $\unicode[STIX]{x1D6E5}_{l}(t)$ as well as the relation between the monodromy zeta function and the Hodge spectrum of the singularity. In the case $n=2$, we give a description of $\unicode[STIX]{x1D6E5}_{C}(t)=\unicode[STIX]{x1D6E5}_{1}(t)$ in terms of the multiplier ideal.

On the topological type of a set of plane valuations with symmetries

AbstractLet be a set of irreducible plane curve singularities. For an action of a finite group G, let be the Alexander polynomial in variables of the algebraic link and let with identical variables in each group. (If , is the monodromy zeta function of the function germ , where is an equation defining the curve C1.) We prove that determines the topological type of the link L. We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincaré series of a set of valuations.

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.

The Milnor fiber of the singularity $$f(x,y) + zg(x,y) = 0$$ f ( x , y ) + z g ( x , y ) = 0

We give a description of the Milnor fiber and the monodromy of a singularity of the form $$f+zg = 0$$ , where f and g define germs of plane curve singularities and have no common components. In particular, this gives a description of the boundary of the Milnor fiber. The description depends only on the topological type of the two plane curve germs defined by f and g. As a corollary, we give a simple formula for the monodromy zeta function and the Euler characteristic of the fiber in terms of an embedded resolution of f and g.

On an Equivariant Version of the Zeta Function of a Transformation

Earlier the authors offered an equivariant version of the classical monodromy zeta function of a \(G\)-invariant function germ with a finite group \(G\) as a power series with the coefficients from the Burnside ring of the group \(G\) tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a \(G\)-equivariant transformation given by W. Luck and J. Rosenberg. Here we use another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta function. This zeta-function is a power series with the coefficients from the Burnside ring itself. We give an A’Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution. Finally we discuss orbifold versions of the Lefschetz number and of the monodromy zeta function corresponding to the two equivariant ones.

An equivariant Poincaré series of filtrations and monodromy zeta functions

We define a new equivariant (with respect to a finite group $$G$$ action) version of the Poincare series of a multi-index filtration as an element of the power series ring $${\widetilde{A}}(G)[[t_1, \ldots , t_r]]$$ for a certain modification $${\widetilde{A}}(G)$$ of the Burnside ring of the group $$G$$ . We give a formula for this Poincare series of a collection of plane valuations in terms of a $$G$$ -resolution of the collection. We show that, for filtrations on the ring of germs of functions in two variables defined by the curve valuations corresponding to the irreducible components of a plane curve singularity defined by a $$G$$ -invariant function germ, in the majority of cases this equivariant Poincare series determines the corresponding equivariant monodromy zeta functions defined earlier.

The monodromy conjecture for plane meromorphic germs

A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function have been introduced by Gusein-Zade, Luengo and Melle-Hernández [‘Zeta functions for germs of meromorphic functions, and Newton diagrams’, Funct. Anal. Appl. 32 (1998)]. In this article, we define the topological zeta function for meromorphic germs and we study its poles in the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.

Monodromy zeta function formula for embedded $\mathbf{Q}$-resolutions

In previous work we have introduced the notion of an embedded \mathbf{Q} -resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities. Here we give a generalization to this setting of N. A’Campo’s formula for the monodromy zeta function of a singularity. Some examples of its application are shown.

On an equivariant analogue of the monodromy zeta function

We offer an equivariant analogue of the monodromy zeta function of a germ invariant with respect to an action of a finite group G as an element of the Grothendieck ring of finite (ℤ×G)-sets. We state equivariant analogues of the Sebastiani-Thom theorem and of the A’Campo formula.

Equivariant Poincaré Series and Monodromy Zeta Functions of Quasihomogeneous Polynomials

In earlier work, the authors described a relation between the Poincaré series and the classical monodromy zeta function corresponding to a quasihomogeneous polynomial. Here we formulate an equivariant version of this relation in terms of the Burnside rings of finite abelian groups and their analogues.

Zeta function of degenerate plane curve singularity

We introduce in this paper a new resolution graph for an isolated complex plane curve singularity and then calculate the monodromy zeta function and the Alexander polynomial for the singularity in terms of this graph.

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.

Saito duality between Burnside rings for invertible polynomials

We give an equivariant version of the Saito duality which can be regarded as a Fourier transformation on Burnside rings. We show that (appropriately defined) reduced equivariant monodromy zeta functions of Berglund–Hübsch dual invertible polynomials are Saito dual to each other with respect to their groups of diagonal symmetries. Moreover, we show that the relation between ‘geometric roots’ of the monodromy zeta functions for some pairs of Berglund–Hübsch dual invertible polynomials described in a previous paper is a particular case of this duality.

Monodromy zeta-function of a polynomial on a complete intersection, and Newton polyhedra

For a generic (polynomial) one-parameter deformation of a complete intersection, there is defined its monodromy zeta-function. We provide explicit formulae for this zeta-function in terms of the corresponding Newton polyhedra in the case the deformation is non-degenerate with respect to its Newton polyhedra. Using this result we obtain the formula for the monodromy zeta-function at the origin of a polynomial on a complete intersection, which is an analog of the Libgober--Sperber theorem.

Monodromy conjecture for nondegenerate surface singularities

We prove the monodromy conjecture for the topological zeta function for all non-degenerate surface singularities. Fundamental in our work is a detailed study of the formula for the zeta function of monodromy by Varchenko and the study of the candidate poles of the topological zeta function yielded by what we call 'B1-facets'. In particular, new cases among the nondegenerate surface singularities for which the monodromy conjecture is proven now, are the non-isolated singularities, the singularities giving rise to a topological zeta function with multiple candidate pole and the ones for which the Newton polyhedron contains a B1-facet.

Milnor fibers over singular toric varieties and nearby cycle sheaves

We apply sheaf-theoretical methods to monodromy zeta functions of Milnor fibrations. Classical formulas due to Kushnirenko, Varchenko and Oka, etc. on polynomials over the complex affine space will be generalized to polynomial functions over any toric variety. Moreover our results enable us to calculate the monodromy zeta functions of any constructible sheaf.

Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves

By using sheaf-theoretical methods such as constructible sheaves, we generalize the formula of Libgober–Sperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions. In particular, some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps will be obtained.

Composition with a two variable function

In [11] and [12], A. Nemethi studied the Milnor fiber and monodromy zeta function of composed functions of the form f(g1, g2) with f a two variable polynomial and g1 and g2 polynomials with distinct sets of variables. The present paper addresses the question of proving similar results for the motivic Milnor fiber introduced by Denef and Loeser, cf. [1],[3],[10],[4]. In fact, Nemethi later considered in [13] the more general situation of a composition f ◦ g : (X,x) → (C, 0) → (C, 0), where g has a reasonable discriminant. Still later, Nemethi and Steenbrink [14] proved similar results at the level of the Hodge spectrum [16],[17],[18], using the theory of mixed Hodge modules. In particular, they were able to compute, under mild assumptions, the Hodge spectrum of composed functions of the form f(g1, g2) without assuming the variables in g1 and g2 are distinct. Their result involves the discriminant of the morphism g = (g1, g2). In a previous paper [7], we computed the motivic Milnor fiber for functions of the form g1 + g 2 when ` is large without assuming the variables in g1 and g2 are distinct. The corresponding result for the Hodge spectrum goes back to M. Saito [15] and is a special case of the results of Nemethi and Steenbrink [14]. So, it seems very natural to search for a full motivic analogue of the results of [14]. At the present time, we are unable to realize this program and we have to limit ourself, as we already mentioned, to the case when g1 and g2 have no variable in common. Already extending our result to the case when one only assumes the discriminant of the morphism g is contained in the coordinate axes seems to require new ideas.