54 research outputs found
Images of analytic map germs, and singular fibrations
For a map germ with target or
with , we address two phenomena which do not occur when : the
image of may be not well-defined as a set germ, and a local fibration near
the origin may not exist. We show how these two phenomena are related, and how
they can be characterised.Comment: revised, to appear in Europ. J. Mat
Local linear Morsifications
The number of Morse points in a Morsification determines the topology of the
Milnor fibre of a holomorphic function germ with isolated singularity. If
has an arbitrary singular locus, then this nice relation to the Milnor
fibre disappears. We show that despite this loss, the numbers of stratified
Morse singularities of a general linear Morsification are effectively
computable in terms of topological invariants of
Limits of tangents and minimality of complex links
AbstractWe show that the complex link of a large class of space germs (X,x0) is characterized by its “simplicity”, among the Milnor fibres of functions with isolated singularity on X. This amounts to the minimality of the Milnor number, whenever this number is defined. Such a phenomenon has been first pointed out in case (X,x0) is an isolated hypersurface singularity, by Teissier (Cycles évanescents, sections planes et conditions de Whitney, in: Singularités à Cargèse 1972, Asterisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285–362)
Bouquet decomposition of the Milnor fibre
AbstractWe consider the Milnor fibre of an isolated singularity ƒ:(X, 0) → (C, 0) on a reduced, Whitney stratified space germ (X, 0) and prove that it is homotopy equivalent to a bouquet of repeated suspensions of complex links of strata.This is a consequence of a “cell attaching” type result which we prove first by investigating the geometric monodromy given by Lê's “carrousel method”.Our results generalise the bouquet theorems of Milnor, Hamm, Lê, Siersma and the handlebody theorem of Lê and Perron
Detecting bifurcation values at infinity of real polynomials
We present a new approach for estimating the set of bifurcation values at
infinity. This yields a significant shrinking of the number of coefficients in
the recent algorithm introduced by Jelonek and Kurdyka for reaching critical
values at infinity by rational arcs
On singular maps with local fibration
We discuss the most general condition under which a singular local tube
fibration exists. We give an application to composition of map germs.Comment: to appear in: Revue Roumaine de Math\'ematiques Pures et
Appliqu\'ees, 2023. A volume dedicated to the memory of Mihnea Coltoi
Polar degree of singular projective hypersurfaces, and vanishing cycles of polynomials
We prove that the polar degree of an arbitrarily singular projective
hypersurface can be decomposed as a sum of non-negative numbers which
represent local vanishing cycles of two different types, namely related to
"special points" of , or related to "non-generic polar curves". We use on
the one hand a non-generic slicing strategy which extends Huh's method, and on
the other hand the study of the vanishing cycles of affine functions. We derive
lower bounds for the polar degree of any singular .Comment: 26
Morse numbers of complex polynomials
To a polynomial function with arbitrary singularities we associate the
number of Morse points in a general linear Morsification .
We produce computable algebraic formulas in terms of invariants of for the
numbers of stratwise Morse trajectories which abut, as , to some point
of or to some point at infinity.Comment: 16
- …