106 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
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
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
The Poincare series of divisorial valuations in the plane defines the topology of the set of divisors
To a plane curve singularity one associates a multi-index filtration on the
ring of germs of functions of two variables defined by the orders of a function
on irreducible components of the curve. The Poincare series of this filtration
turnes out to coincide with the Alexander polynomial of the curve germ. For a
finite set of divisorial valuations on the ring corresponding to some
components of the exceptional divisor of a modification of the plane, in a
previous paper there was obtained a formula for the Poincare series of the
corresponding multi-index filtration similar to the one associated to plane
germs. Here we show that the Poincare series of a set of divisorial valuations
on the ring of germs of functions of two variables defines "the topology of the
set of the divisors" in the sense that it defines the minimal resolution of
this set up to combinatorial equivalence. For the plane curve singularity case,
we also give a somewhat simpler proof of the statement by Yamamoto which proves
that the Alexander polynomial is equivalent to the embedded topology
The Alexander polynomial of a plane curve singularity via the ring of functions on it
We prove two formulae which express the Alexander polynomial of
several variables of a plane curve singularity in terms of the ring of germs of analytic functions on the curve. One of them expresses
in terms of dimensions of some factors corresponding to a
(multi-indexed) filtration on the ring . The other one gives the
coefficients of the Alexander polynomial as Euler characteristics of
some explicitly described spaces (complements to arrangements of projective
hyperplanes). The final version of this article will be published in the Duke
Mathematical Journal
The Alexander polynomial of a plane curve singularity and the ring of functions on it
We give two formulae which express the Alexander polynomial of
several variables of a plane curve singularity in terms of the ring of germs of analytic functions on the curve. One of them expresses
in terms of dimensions of some factorspaces corresponding to a
(multi-indexed) filtration on the ring . The other one gives the
coefficients of the Alexander polynomial as Euler characteristics of
some explicitly described spaces (complements to arrangements of projective
hyperplanes).Comment: 5 pages, LaTe
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