Stein Manifolds and Holomorphic Mappings

1. Front Matter

   Pages I-XV

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2. Stein Manifolds

   1. Front Matter

      Pages 1-2

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   2. Preliminaries

      • Franc Forstnerič

      Pages 3-43

   3. Stein Manifolds

      • Franc Forstnerič

      Pages 45-64

   4. Stein Neighborhoods and Approximation

      • Franc Forstnerič

      Pages 65-106

   5. Automorphisms of Complex Euclidean Spaces

      • Franc Forstnerič

      Pages 107-203

3. Oka Theory

   1. Front Matter

      Pages 205-205

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   2. Oka Manifolds

      • Franc Forstnerič

      Pages 207-262

   3. Elliptic Complex Geometry and Oka Theory

      • Franc Forstnerič

      Pages 263-317

   4. Flexibility Properties of Complex Manifolds and Holomorphic Maps

      • Franc Forstnerič

      Pages 319-349

4. Applications

   1. Front Matter

      Pages 351-351

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   2. Applications of Oka Theory and Its Methods

      • Franc Forstnerič

      Pages 353-402

   3. Embeddings, Immersions and Submersions

      • Franc Forstnerič

      Pages 403-476

   4. Topological Methods in Stein Geometry

      • Franc Forstnerič

      Pages 477-531

5. Back Matter

   Pages 533-562

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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.

• Stein manifold
• Oka manifold
• elliptic manifold
• holomorphic map
• holomorphic automorphism
• holomorphic fibre bundle
• Oka-Grauert principle
• homotopy principle
• holomorphic spray
• homotopy equivalence
• Stein spaces
• Stein neighborhoods
• Oka theory applications
• complex manifolds flexibility properties
• holomorphic maps flexibility properties
• Stein geometry topological methods
• 32E10, 32H02, 32L05, 32M12, 32M17, 14M17, 58D15

• Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

  Franc Forstnerič

Franc Forstneric has published more than a hundred research and survey papers in complex analysis and geometry, including several in leading mathematical journals such as the Annals of Math., Acta Math., Inventiones Math., Duke Math. J., J. Eur. Math. Soc., Amer. J. Math., and others.

He held long term teaching and research positions at the

University of Wisconsin-Madison (Madison, USA),

Centre for Advanced Study (Oslo, Norway),

Institut Mittag-Leffler (Stockholm, Sweden),

Max Planck Institute (Bonn, Germany),

as well as visiting positions at more than ten other institutions. He was an invited speaker at over a hundred international conferences and workshops.

Since 2000 he is a Professor of Mathematics at the University of Ljubljana and is a member of the Academy of Sciences and Arts of the Republic of Slovenia.

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