121 research outputs found
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 of germs of functions on a germ 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 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 action)
version of the Poincar\'e series of a multi-index filtration as an element of
the power series ring for a certain
modification of the Burnside ring of the group . We
give a formula for this Poincar\'e series of a collection of plane valuations
in terms of a -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 -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 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 act on the complex plane . 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 defined by components of a modification of the
complex plane at the origin or by branches of a -invariant
plane curve singularity . 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
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