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.
Daniel Barlet studied mathematics at the ENS Ulm (Paris) from 1966 to 1970. After his graduation he became assistant professor at the University of Paris VII, where he defended his thèse d’État in December 1974. From 1976 to 2011 he was professor at the University of Nancy I, which is today a part of the University of Lorraine. He was president of the French Mathematical Society (92/94) and was elected as a senior member of the Institut Universitaire de France (Analysis and Complex Geometry) from 1998 to 2003 and his chair was renewed for five years in 2003. Since 2011 he has been professor emeritus at the Institute of Elie Cartan at the University of Lorraine.
Jón Magnússon finished a B.Sc. in mathematics at the University of Iceland in 1976 and obtained a doctoral degree in mathematics from the University of Paris VII (Jussieu) in 1981. Since then he has been working, first as a research mathematician and later as a professor, at the University of Iceland. In 2023 he became a professor emeritus at the University of Iceland. His research interests are mainly in cycle space theory.
