Jón Magnússon – författare

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.

This book is the second volume of a work on complex analytic cycles and the results, stated without proof in the first volume, are proved here. It begins with the construction of the reduced complex space formed by all compact cycles of a given complex space. Following this construction the main subjects of the book are:

• Fundamental class of a cycle and relative fundamental class of an analytic family of cycles

• Intersection theory with parameters on complex manifolds and more generally on nearly smooth complex spaces

• Holomorphic currents on reduced complex spaces

• Chow varieties and cycle spaces of quasi-projective complex spaces

• Natural morphism from the Douady space to the cycle space

• Holomorphic convexity in cycle spaces and integration of $\bar{partial}$-cohomology classes on cycles

• Strong Kählerianity of cycle spaces of Kähler manifolds

• Numerous important applications of cycle space theory

Preliminaries needed in the book in addition to the material of the first volume, for instance sheaf cohomology with support, are explained in detail, making this two-volume work quite self-contained. The French version of the present book was published in 2020 by the French Mathematical Society in the series Cours Spécialisés and during the translation process the authors have in many ways improved the original version.