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2017 | Buch

Kapitel lesen Erstes Kapitel lesen

Stein Manifolds and Holomorphic Mappings

The Homotopy Principle in Complex Analysis

verfasst von: Prof. Franc Forstnerič

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Stein Manifolds

Frontmatter

Chapter 1. Preliminaries

Abstract

In this preliminary chapter we review the basic notions concerning complex manifolds, complex spaces with singularities, holomorphic mappings, holomorphic fibre bundles and especially vector bundles, the tangent and cotangent bundle of a complex manifold, differential forms, de Rham and Dolbeault cohomology, plurisubharmonic functions, the Levi form, vector fields and their flows. We also recall a basic form of Gromov’s homotopy principle for first order partial differential relations that are ample in the coordinate directions.

Franc Forstnerič

Chapter 2. Stein Manifolds

Abstract

This chapter contains a brief survey of the classical theory of Stein manifolds and Stein spaces. After a historical introduction we recall the Cartan-Thullen characterization of domains of holomorphy by holomorphic convexity, the notions of Hartogs and Levi pseudoconvexity, the Levi problem, the basic characterizations of Stein manifolds and Stein spaces, Cartan-Serre Theorems A and B and their immediate applications, holomorphic embedding theorems for Stein manifolds, and solvability of the nonhomogeneous Cauchy-Riemann equations. We include a proof of a parametric version of the Cartan-Oka-Weil theorem for sections of holomorphic vector bundles on reduced Stein spaces. This result plays an important role in the development of Oka theory in Chaps. 5 and 6.

Franc Forstnerič

Chapter 3. Stein Neighborhoods and Approximation

Abstract

A major part of this chapter is devoted to the construction of open Stein neighborhoods of certain types of sets in complex spaces. Highlights include Siu’s theorem on the existence of Stein neighborhoods of Stein subvarieties and various generalizations, the Docquier-Grauert type theorems on the existence of holomorphic retractions onto Stein submanifolds, and the construction of Stein neighborhoods of compact holomorphically convex sets with attached totally real submanifolds. With these results in hand, we prove certain extension and approximation theorems for holomorphic mappings and for sections of holomorphic submersions. We also analyze the geometry of Morse critical points of strongly plurisubharmonic and \(q\)-convex functions, and we describe the topological structure of Stein spaces and of \(q\)-complete complex spaces.

Franc Forstnerič

Chapter 4. Automorphisms of Complex Euclidean Spaces

In this chapter we study holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with big holomorphic automorphism groups. The main focus is the Andersén-Lempert theory and its generalization to complex manifolds enjoying Varolin’s density property. This subject is closely intertwined with Oka theory and furnishes important examples of Oka manifolds. Our knowledge of the class of Stein manifolds with the density property has advanced considerably since the first edition of this book was published, and we describe most of these new developments. Applications presented in the chapter include the construction of nonstraightenable embedded complex lines, of twisted proper holomorphic embeddings of Stein spaces into Euclidean spaces, of nonlinearizable periodic holomorphic automorphisms of \({\mathbb {C}}^{n}\), of non-Runge Fatou-Bieberbach domains, and of big families of pairwise distinct long \({\mathbb {C}}^{n}\)’s which do not admit any nonconstant holomorphic or plurisubharmonic functions.

Franc Forstnerič

Oka Theory

Frontmatter

Chapter 5. Oka Manifolds

Abstract

This is the first of the two core chapters of the book. We begin with a historical introduction to the Oka-Grauert principle which says that the topological classification of principal fibre bundles over Stein spaces agrees with the holomorphic classification. Next we describe a reduction to the problem of deforming families of continuous sections to families of holomorphic sections in principal fibre bundles over Stein spaces. This naturally leads to the theory of Oka manifolds. A complex manifold \(Y\) is said to be an Oka manifold if every holomorphic map from any convex set in a Euclidean space \({\mathbb {C}}^{n}\) can be approximated by entire maps \({\mathbb {C}}^{n}\to Y\). The main result is that sections of any stratified holomorphic fibre bundle with Oka fibres over a reduced Stein space enjoy all forms of the Oka principle. Since every complex homogeneous manifold is Oka, this generalizes the classical Oka-Grauert theory. We give a complete proof, proceeding in steps from the simplest to the most general case. We also discuss properties and examples of Oka manifold and give several nontrivially equivalent characterizations of this class.

Franc Forstnerič

Chapter 6. Elliptic Complex Geometry and Oka Theory

Abstract

In this second core chapter we extend the Oka principle to the more general setting of certain non-locally trivial holomorphic submersions \(Z\to X\) over a reduced Stein space \(X\). The main result is a theorem of Gromov from 1989 and several of its generalizations. Roughly stated, it says that the existence of a finite dominating family of holomorphic fibre sprays on \(Z\) over each sufficiently small open subset of \(X\) (a submersion with this property is said to be subelliptic) implies all forms of the Oka principle for sections \(X\to Z\). The analogous result is proved for stratified subelliptic submersions. We also give an axiomatic version of this result, where the necessary local property of the submersion concerns the possibility of approximating homotopies of holomorphic sections over small open sets in the base \(X\). We give a relative Oka principle for sections of certain branched holomorphic maps, and we prove a Runge type approximation theorem for algebraic maps from affine algebraic varieties to a certain class of algebraic manifolds.

Franc Forstnerič

Chapter 7. Flexibility Properties of Complex Manifolds and Holomorphic Maps

Abstract

This chapter is new in the current edition and focuses on recent developments. We begin by comparing the Oka property with other standard holomorphic flexibility properties of complex manifolds that have been studied in the literature. We introduce the class of stratified Oka manifolds and show that every such manifold is strongly dominable by the complex Euclidean space of the same dimension. Next we describe what we know about which compact complex surfaces are Oka. We also introduce and study the class of Oka maps. In a section contributed by Finnur Lárusson we give a homotopy-theoretic viewpoint of modern Oka theory. We conclude by discussing miscellaneous recent results and collecting open problems.

Franc Forstnerič

Applications

Frontmatter

Chapter 8. Applications of Oka Theory and Its Methods

Abstract

In this chapter we apply the methods and results of Oka theory to a variety of problems in Stein geometry. In particular, we discuss the structure of holomorphic vector bundles and their homomorphisms over Stein spaces, find the minimal number of generators of coherent analytic sheaves over Stein spaces, consider the problem of complete intersections and, more generally, of elimination of intersections of holomorphic maps with complex subvarieties, present the solution of the holomorphic Vaserstein problem, discuss transversality theorems for holomorphic and algebraic maps from Stein manifolds, and obtain estimates on the dimension of singularity (degeneration) sets of generic holomorphic maps from Stein manifolds to Oka manifolds. In the last part of the chapter we further develop the method of local holomorphic sprays of maps from strongly pseudoconvex Stein domains. We apply this technique to prove an up-to-the-boundary version of the Oka principle on such domains, and we establish a Banach manifold structure theorem for certain spaces of holomorphic maps from strongly pseudoconvex domains to arbitrary complex manifolds.

Franc Forstnerič

Chapter 9. Embeddings, Immersions and Submersions

In this chapter we study holomorphic immersion, embeddings, and submersions of Stein manifolds to Euclidean spaces and to certain other complex manifolds. We begin by exploring the homotopy principle for totally real immersions and embeddings. The main results of the chapter include the optimal embedding and immersion theorems for Stein manifolds to Euclidean spaces, the Oka principle for holomorphic immersions of Stein manifolds to Euclidean spaces, the construction of proper holomorphic maps of strongly pseudoconvex Stein domains to \(q\)-convex manifolds, the construction of proper holomorphic embeddings of certain open Riemann surfaces into the affine plane \({\mathbb {C}}^{2}\), the construction of noncritical holomorphic functions on Stein manifolds and Stein spaces, the homotopy principle for holomorphic submersions from Stein manifolds, and the construction of nonsingular holomorphic foliations on Stein manifolds. An important technical result proved in this chapter is a splitting lemma for biholomorphic maps close to the identity on a Cartan pair.

Franc Forstnerič

Chapter 10. Topological Methods in Stein Geometry

Abstract

In this chapter we introduce more advanced topological methods to the study of geometric problems on Stein manifolds and to Oka theory. We begin by considering complex points of smooth real surfaces in complex surfaces. After proving the Lai formulae and the Eliashberg-Harlamov cancellation theorem, we explore connections between the generalized adjunction inequality and the existence of embedded or immersed surfaces with a Stein neighborhood basis in a given complex surface. We then show how the Seiberg-Witten theory bears upon these questions by arguments similar to those leading to the proof of the generalized Thom conjecture. In the second part we present the Eliashberg-Gompf construction of integrable Stein structures on almost complex manifolds with a suitable handlebody decomposition, and we prove the soft Oka principle to the effect that every continuous map from a Stein manifold \(X\) to an arbitrary complex manifold \(Y\) is homotopic to a holomorphic map provided that we allow homotopic deformations of the Stein structure on \(X\) and, in real dimension four, also a change of the underlying smooth structure on \(X\).

Franc Forstnerič

Backmatter

Titel: Stein Manifolds and Holomorphic Mappings
verfasst von: Prof. Franc Forstnerič
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-61058-0
Print ISBN: 978-3-319-61057-3
DOI: https://doi.org/10.1007/978-3-319-61058-0