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Stein Manifolds and Holomorphic Mappings [electronic resource] : The Homotopy Principle in Complex Analysis / by Franc Forstnerič.
2nd ed. 2017.
Cham : Springer International Publishing : Imprint: Springer, 2017.
1 online resource (XV, 562 p. 29 illus., 1 illus. in color.)
Functions of complex variables
Global analysis (Mathematics).
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 56
[More in this series]
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 0071-1136 ; 56
This book, now in a carefully revised second edition, provides an up-to-date account of Oka theory, including the classical Oka-Grauert theory and the wide array of applications to the geometry of Stein manifolds. Oka theory is the field of complex analysis dealing with global problems on Stein manifolds which admit analytic solutions in the absence of topological obstructions. The exposition in the present volume focuses on the notion of an Oka manifold introduced by the author in 2009. It explores connections with elliptic complex geometry initiated by Gromov in 1989, with the Andersén-Lempert theory of holomorphic automorphisms of complex Euclidean spaces and of Stein manifolds with the density property, and with topological methods such as homotopy theory and the Seiberg-Witten theory. Researchers and graduate students interested in the homotopy principle in complex analysis will find this book particularly useful. It is currently the only work that offers a comprehensive introduction to both the Oka theory and the theory of holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with large automorphism groups.
Includes bibliographical references and index.
Part I Stein Manifolds -- 1 Preliminaries -- 2 Stein Manifolds -- 3 Stein Neighborhoods and Approximation -- 4 Automorphisms of Complex Euclidean Spaces -- Part II Oka Theory -- 5 Oka Manifolds -- 6 Elliptic Complex Geometry and Oka Theory -- 7 Flexibility Properties of Complex Manifolds and Holomorphic Maps -- Part III Applications -- 8 Applications of Oka Theory and its Methods -- 9 Embeddings, Immersions and Submersions -- 10 Topological Methods in Stein Geometry -- References -- Index.
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Stein manifolds and holomorphic mappings : the homotopy principle in complex analysis / Franc Forstnerič.