This book highlights the use of non-compact analytic cycles in complex geometry. The main focus is on analytic families of cycles of finite type, in other words, cycles which have only finitely many irreducible components. It is shown how the space of all cycles of finite type in a given complex space, endowed with a weak analytic structure, can be used in many ways as the reduced complex space of all compact cycles in the given space. Several illustrative and enlightening examples are provided, as well as applications, giving life to the theory. The exposition includes a characterization of quasi-proper holomorphic maps which admit a geometric flattening, a proof of an existence theorem for meromorphic quotients with respect to a large class of analytic equivalence relations, and a generalization of the Stein factorization to a variety of holomorphic maps. In addition, a study is made of the behavior of analytic families of finite type cycles when they are restricted to Zariski open subsets and extended across analytic subsets.

Aimed at researchers and graduate students with an interest in complex or algebraic geometry, the book is adequately self-contained, the basic notions are explained and suitable references are given for auxiliary results that are used in the text.

About the Author

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 these d'etat 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 (Complex Analysis and Complex Geometry) from 1998 to 2003. His chair was renewed for a further five years in 2003. Since 2011 he has been professor emeritus at the Institute of Elie Cartan at the University of Lorraine.

J�n Ing�lfur Magn�sson obtained a B.Sc. in mathematics at the University of Iceland in 1976 and 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.

The authors wrote a two volume work on analytic cycles in complex spaces which was published by the French Mathematical Society (Cours Sp�cialis�s 22 and 27) in 2014 and 2020. An improved English version of the first volume was published by Springer (Grundlehren der mathematischen Wissenschaften 356) in 2019 and an English version of the second volume will soon be published in the same series.

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