### Geometry of Foliations

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Examples : The circle S 1 is decomposable.

### Journal of Differential Geometry

The Knaster Continuum or bucket handle is indecomposable. This is one-half of a Smale Horseshoe. The 2-solenoid over S 1 is a branched double Steven Hurder.

Index theory and non-commutative geometry on foliated manifolds Yuri A. References Publications referenced by this paper. Coarse cohomology and index theory on complete Riemannian manifolds John Roe. Expansion growth of foliations Shinji Egashira. Singular Riemannian Foliations Pierre Molino.

Baum , Alain Connes. The graph of a foliation Horst Elmar Winkelnkemper. Related Papers. It introduces the notion of the landscape of a singularity as the right setting for equisingularity problems. This survey paper is focused on discussing the main facts from the orbital formal rigidity phenomenon for germs of holomorphic and real analytic vector fields in the complex and real planes, exploring their similar and different properties.

In this paper we study holomorphic foliations with singularities having a homogeneous transverse structure of projective model i.

## Geometry of Foliations by Philippe Tondeur

The main case occurs, as we shall see, when the analytic set is invariant by the foliation. We address both, the local and the global cases. Our focus is the extension of the structure in a suitable sense. After performing a characterization of the existence of the structure in terms of suitable triples of differential forms, we consider the problem of extension of such structures to the analytic invariant set for germs of foliations and for foliations in complex projective spaces.

Basic examples of this situation are given by logarithmic foliations and Riccati foliations. We also study the holonomy of such invariant sets, as a consequence of a strict link between this holonomy and the monodromy of a projective structure. These holonomy groups are proved to be solvable. Our final aim is the classification of such object under some mild conditions on the singularities they exhibit.

In this work we perform this classification in the case where the singularities of the foliation are supposed to be non-dicritical and non-degenerate more precisely, generalized curves. This case, we will see, corresponds to the transversely affine case and therefore to the class of logarithmic foliations. The more general case, which has to do with Riccati foliations, is dealt with by some extension results we prove and evoking results from Loray-Touzet-Vitorio.

We provide a reasonably detailed introduction to the ICSS, including some low-dimensional examples of its use. The paper is partly expository. We also give the 2-colored versions of all these results, when the submanifold separates the ambient manifold into two parts. We describe how the study of the singularities of height and distance squared functions on submanifolds of Euclidean space, combined with adequate topological and geometrical tools, shows to be useful to obtain global geometrical properties. Irreducible complex plane curve germs with the same characteristic exponents form an equisingularity class.

In this paper we determine the Zariski invariants that characterize the general polar of a general member of such an equisingularity class. In this article, we introduce a notion of non-degeneracy, with respect to certain Newton polyhedra, for rational functions over non-Archimedean local fields of arbitrary characteristic.

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In contrast with the classical local zeta functions, the meromorphic continuations of zeta functions for rational functions have poles with positive and negative real parts. We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject. We discuss some open problems and questions related with five different topics in complex singularities.

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These are: i Topological and holomorphic ranks of an isolated singularity germ and the Zariski-Lipman conjecture; ii Graph manifolds and links of surface singularities. These are all topics on which I have been interested for a long time.

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Under mild conditions we describe a complete set of analytic invariants characterizing foliations with quasi-homogeneous separatrices. Further, we give the full moduli space of quasi-homogeneous plane curves. This paper has an expository character in order to make it accessible also to non-specialists.