scholarly journals Topological equivalence of holomorphic foliation germs of rank $1$ with isolated singularity in the Poincaré domain

2019 ◽  
Vol 69 (2) ◽  
pp. 561-590
Author(s):  
Thomas Eckl ◽  
Michael Lönne
Keyword(s):  
Holomorphic Foliation ◽  
Topological Equivalence ◽  
Isolated Singularity

On the versal deformation of a complex space with an isolated singularity

1972 ◽  
Vol 196 (1) ◽  
pp. 23-29 ◽  
Author(s):  
Arnold Kas ◽  
Michael Schlessinger
Keyword(s):  
Complex Space ◽  
Versal Deformation ◽  
Isolated Singularity

scholarly journals The jump of the Milnor number in the X 9 singularity class

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Szymon Brzostowski ◽  
Tadeusz Krasiński
Keyword(s):  
Milnor Number ◽  
Isolated Singularity ◽  
Milnor Numbers

AbstractThe jump of the Milnor number of an isolated singularity f 0 is the minimal non-zero difference between the Milnor numbers of f 0 and one of its deformations (f s). We prove that for the singularities in the X 9 singularity class their jumps are equal to 2.

scholarly journals Generic pseudogroups on and the topology of leaves

2013 ◽  
Vol 149 (8) ◽  
pp. 1401-1430 ◽  
Author(s):  
J.-F. Mattei ◽  
J. C. Rebelo ◽  
H. Reis
Keyword(s):  
Invariant Curve ◽  
Holomorphic Foliation ◽  
Simply Connected

AbstractWe show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

INTERTWINED BASINS OF ATTRACTION

2012 ◽  
Vol 22 (06) ◽  
pp. 1250130
Author(s):  
CHANGMING DING
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Basins Of Attraction ◽  
General Definition ◽  
Sufficient Condition ◽  
Topological Equivalence ◽  
Definition Of

This paper deals with intertwined basins of attraction for dynamical systems in a metric space. After giving a general definition of intertwining property, which is preserved by a topological equivalence between dynamical systems, we present a sufficient condition to guarantee the existence of intertwined basins for dynamical systems in ℝn.

scholarly journals On the geometry of Poincaré's problem for one-dimensional projective foliations

2001 ◽  
Vol 73 (4) ◽  
pp. 475-482 ◽  
Author(s):  
MARCIO G. SOARES
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Projective Variety ◽  
Holomorphic Foliation ◽  
One Dimensional ◽  
Complex Projective Variety ◽  
Geometric Objects ◽  
Complex Projective

We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a one-dimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation.

scholarly journals Affine varieties, singularities and the growth rate of wrapped Floer cohomology

2018 ◽  
Vol 10 (03) ◽  
pp. 493-530
Author(s):  
Mark McLean
Keyword(s):  
Growth Rate ◽  
Cotangent Bundle ◽  
Betti Numbers ◽  
Symplectic Manifolds ◽  
Isolated Singularity ◽  
Contact Manifolds ◽  
Floer Cohomology ◽  
Complex Singularities ◽  
Smooth Affine ◽  
Simply Connected

In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology. Using this invariant we show that if [Formula: see text] is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationally elliptic and so it must have certain bounds on its Betti numbers.

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