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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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminary Material

Abstract
The first paragraph of this chapter is dedicated to a brief, elementary discussion of holomorphic maps from open sets in Banach spaces with values in Banach spaces. For this, only knowledge of the basic properties of holomorphic functions on open sets in \({\mathbb {C}}\) is assumed. It should be noted that this material will be used in a rather simple way in an infinite dimensional setting in Section 1.4. The material on holomorphic maps is followed by a quick introduction to the notions of complex manifold and holomorphic vector bundle. In particular we introduce projective spaces and Grassmannians.
Daniel Barlet, Jón Magnússon

Chapter 2. Multigraphs and Reduced Complex Spaces

Abstract
This chapter begins with a detailed discussion of the notion of areduced multigraph which corresponds in the classical literature to the notion of anembedded ramified cover. We will systematically use multigraphs to serve as the local models for reduced, pure-dimensional complex spaces.
Daniel Barlet, Jón Magnússon

Chapter 3. Analysis and Geometry on a Reduced Complex Space

Abstract
This chapter focuses on three fundamental tools for working with reduced complex spaces.
Daniel Barlet, Jón Magnússon

Chapter 4. Families of Cycles in Complex Geometry

Abstract
When considering the roots of a monic polynomial of degree k which depends analytically on a parameter, one is led to consider not only k-tuples of pairwise distinct points in \(\mathbb {C}\), but also k-tuples containing repeated points which correspond to multiple roots which can appear for certain values of the parameter. This led us to study \( \operatorname {\mathrm {Sym}}^k(\mathbb {C})\) and then \( \operatorname {\mathrm {Sym}}^k(\mathbb {C}^p)\) in Chapter 1.
Daniel Barlet, Jón Magnússon

Backmatter

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