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  • Author: Bruno Scárdua x
  • Mathematics x
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Abstract

We study the classification of polynomial vector fields in two complex variables under the hypotheses that the singularities are isolated and the flow is complete. Normal forms are obtained for the case the generic orbit is diffeomorphic to ℂ. For the case the generic orbit is diffeomorphic to ℂ ∖ {0} and there is an affine singularity we classify the linear part of the vector field and prove the existence of entire linearization or first integral.

Abstract

The geometric theory of foliations had its origins in the classical work of Ehresmann [30, 31] and Reeb [65, 66]. It has a wide diversity of applications and variety of technics, imported from various fields of mathematics as topology, geometry, analysis and dynamical systems. Geometric theory of foliations has been giving a key contribution to the comprehension of various problems in mathematics. We mention for instance the study and classification of real differentiable three-manifolds. The classical real (nonsingular) framework has some key outcomes. For example, we have the theorems of stability (local and global), due to Reeb. Another example is the theorem of Novikov on the existence of a compact torus leaf for a foliation of codimension one in the threesphere S3, and the rank theorem of Lima, on the rank of the three-sphere S3. The notion of a complex foliation (holomorphic foliation) in turn is more recent, though it is already present in spirit in the work of Painlevé [63, 64]. The field has had an intense development in the last decades mostly due to the successful use of modern techniques of complex geometry and several complex variables. Much of the research in complex foliations is centered on local aspects of the theory, for example, the study of singularities of holomorphic foliations. Such a study is already a very hard work and has been very useful in general. However, some global aspects of the theory also deserve special attention. The algebraic case is a kind of “compact case” in this singular framework. In the first chapter we introduce the concept of a holomorphic foliation also in the case of foliations with singularities and we present several examples and classical constructions.

Abstract

In this chapter we study foliations in complex projective spaces. Thanks to its algebraic nature and other geometric-analytic properties, these spaces constitute an important and natural ambient for the study of these objects. This study is rich enough to concentrate on.