Radial index and Euler obstruction of a 1-form on a singular variety

A notion of the radial index of an isolated singular point of a 1-form on a singular (real or complex) variety is discussed. For the differential of a function it is related to the Euler characteristic of the Milnor fibre of the function. A connection between the radial index and the local Euler obstruction of a 1-form is described. This gives an expression for the local Euler obstruction of the differential of a function in terms of Euler characteristics of some Milnor fibres

On the arc filtration for the singularities of Arnold's lists

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincar\'e series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincar\'e series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency relations. Besides that we give a general formula for the Poincar\'e series of the arc filtration for isolated surface singularities which are stabilizations of plane curve ones

Index of a singular point of a vector field or of a 1-form on an orbifold

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier there was defined a notion of the universal Euler characteristic of an orbifold. It takes values in a ring R, as an abelian group freely generated by the generators, corresponding to the isomorphism classes of finite groups. Here we define the universal index of an isolated singular point of a vector field or of a 1-form on an orbifold as an element of the ring R. For this index, an analogue of the Poincar\'e-Hopf theorem holds

Verlinde Algebras and the Intersection Form on Vanishing Cycles

We prove Zuber's conjecture establishing connections of the fusion rules of the $su(N)_k$ WZW model of conformal field theory and the intersection form on vanishing cycles of the associated fusion potential.Comment: latex fil

Universal abelian covers of rational surface singularities and multi-index filtrations

In previous papers, there were computed the Poincare series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincare series were written as the integer parts of certain fractional power series, an interpretation of whom was not given. Here we show that, up to a simple change of variables, these fractional power series are specializations of the equivariant Poincare series for filtrations on the ring of germs of functions on the universal abelian cover of the surface singularity. We compute these equivariant Poincare series

An equivariant Poincar\'e series of filtrations and monodromy zeta functions

We define a new equivariant (with respect to a finite group $G$ action) version of the Poincar\'e 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 Poincar\'e 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 Poincar\'e series determines the corresponding equivariant monodromy zeta functions defined earlier

On Poincar\'e series of filtrations

In this survey one discusses the notion of the Poincar\'e series of multi-index filtrations, an alternative approach to the definition, a method of computation of the Poincar\'e series based on the notion of integration with respect to the Euler characteristic (or rather on an infinite-dimensional version of it), generalizations of the notion of the multi-variable Poincar\'e series based on the notion of the motivic integration, and relations of the latter ones with some zeta functions over finite fields and with generating series of Heegaard-Floer homologies of algebraic links

Integration with respect to Euler characteristic over the projectivization of the space of functions and the Alexander polynomial of a plane curve singularity

We discuss a notion of integration with respect to the Euler characteristic in the projectivization \P{\cal O}_{\C^n,0} of the ring {\cal O}_{\C^n,0} of germs of functions on $C^n$ and show that the Alexander polynomial and the zeta-function of a plane curve singularity can be expressed as certain integrals over \P{\cal O}_{\C^2,0} with respect to the Euler characteristic

On Poincare series of filtrations on equivariant functions of two variables

Let a finite group $G$ act on the complex plane $({\Bbb C}^2, 0)$. We consider multi-index filtrations on the spaces of germs of holomorphic functions of two variables equivariant with respect to 1-dimensional representations of the group $G$ defined by components of a modification of the complex plane ${\Bbb C}^2$ at the origin or by branches of a $G$-invariant plane curve singularity $(C,0)\subset({\Bbb C}^2,0)$. We give formulae for the Poincare series of these filtrations. In particular, this gives a new method to obtain the Poincare series of analogous filtrations on the rings of germs of functions on quotient surface singularities

Integrals with respect to the Euler characteristic over spaces of functions and the Alexander polynomial

We discuss some formulae which express the Alexander polynomial (and thus the zeta-function of the classical monodromy transformation) of a plane curve singularity in terms of the ring of functions on the curve. One of them describes the coefficients of the Alexander polynomial or of zeta-function of the monodromy transformation as Euler characteristics of some explicitly constructed spaces. For the Alexander polynomial these spaces are complements to arrangements of projective hyperplanes in projective spaces. For the zeta-function they are disjoint unions of such spaces. Under the influence of a result by J.Denef and F.Loeser it was understood that this results are connected with the notion of the motivic integration or rather with its version (in some sense a dual one) where the space of arcs is substituted by the space of functions. The aim of this paper is to discuss the notion of the integral with respect to the Euler characteristics (or with respect to the generalized Euler characteristic) over the space of functions (or over its projectivization) and its connection with the formulae for the coefficients of the Alexander polynomial and of the zeta-function of the monodromy transformation as Euler characteristics of some spaces. The paper will be published in Proceedings of the Steklov Mathematical Institute