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The book consists of a presentation from scratch of cycle space methodology in complex geometry. Applications in various contexts are given. A significant portion of the book is devoted to material which is important in the general area of complex analysis. In this regard, a geometric approach is used to obtain fundamental results such as the local parameterization theorem, Lelong' s Theorem and Remmert's direct image theorem. Methods involving cycle spaces have been used in complex geometry for some forty years. The purpose of the book is to systematically explain these methods in a way which is accessible to graduate students in mathematics as well as to research mathematicians. After the background material which is presented in the initial chapters, families of cycles are treated in the last most important part of the book. Their topological aspects are developed in a systematic way and some basic, important applications of analytic families of cycles are given. The construction of the cycle space as a complex space, along with numerous important applications, is given in the second volume. The present book is a translation of the French version that was published in 2014 by the French Mathematical Society.

Functions of complex variables. --- Projective geometry. --- Several Complex Variables and Analytic Spaces. --- Projective Geometry. --- Complex variables --- Elliptic functions --- Functions of real variables --- Projective geometry --- Geometry, Modern --- Geometry, Algebraic. --- Algebraic geometry --- Geometry

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"L'objet de cet ouvrage est, partant de connaissances standards acquises au niveau d'un master de mathématique, d'exposer un certain nombre de résultats de base de la géométrie complexe d'un point de vue résolument géométrique. Les outils introduits sont alors utilisés pour décrire les familles analytiques de cycles dans un espace complexe donné, pour montrer les premières propriétés de stabilité de cette notion et pour donner quelques applications simples mais fondamentales de ces résultats. Ceci sera approfondi et complété dans le second volume." Editeur

Espaces analytiques