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Geometric methods in complex analysis. (English) Zbl 1028.32011
Casacuberta, Carles (ed.) et al., 3rd European congress of mathematics (ECM), Barcelona, Spain, July 10-14, 2000. Volume II. Basel: Birkhäuser. Prog. Math. 202, 55-64 (2001).
This paper is a write-up of a survey talk on some geometrical/topological aspects of several complex variables.
The theme is that Stein manifolds can be used profitably to study envelopes of holomorphy, holomorphic, polynomial or rational hulls of compact sets, and Runge-type approximation. In doing so geometric topology is of great service: (a) One can use Morse functions to describe Stein manifolds as being diffeomorphic to interiors of possibly infinite handlebodies which lack handles of index higher than the complex dimension of the Stein manifold in question (Andreotti-Frankel, 1959). This implies that envelopes of holomorphy “fill in” cycles of dimension higher than the middle dimension, and in Runge manifolds even the middle dimensional cycles vanish. (b) Eliashberg showed in 1990 that the Andreotti-Frankel theorem has a converse in dimensions three and above in the sense of differential topology. (c) Gompf in 1996 completed the picture by providing a converse to the Andreotti-Frankel theorem in dimension two, and obtained non-smooth homeomorphisms.
Thus the remaining general theorems and interesting phenomena about analytic continuation and hulls live in complex dimension 2. Here one has: (d) A theorem of Forstnerič in 1992 that compact oriented real surfaces of genus \(1\) or higher embed in \({\mathbb C}^2\) smoothly with arbitrarily small Stein neighborhoods. (e) The author showed [S. Yu. Nemirovskij, Dokl. Akad. Nauk 362, 442-444 (1998; Zbl 0959.32034)] that to spheres (genus zero) the above theorem of Forstnerič does not generalize.
The main point of the paper is that the following so-called adjunction inequality can be applied decisively to bear upon problems of analytic continuation, hulls, and embeddings of real surfaces into complex ones with Stein neighborhood bases. We quote two results of the author:
Theorem 4.3. (Adjunction Inequality) Let \(\Sigma\subset X\) be a smoothly embedded closed oriented real surface of genus \(g(\Sigma)\) in a Stein complex surface \(X\). If \(\Sigma\) is not a null-homologous \(2\)-sphere in \(X\), then the inequality \([\Sigma]\cdot[\Sigma]+|c_1(X)\cdot[\Sigma]|\leq 2g(\Sigma)-2\) holds, where the first summand is the self-intersection number of \(\Sigma\) in \(X\), and \(c_1(X)\) is the first Chern class of \(X\) [see S. Nemirovskij, Usp. Mat. Nauk 54, No. 4, 47-74); translation in Russ. Math. Surv. 54, No. 4, 729-752 (1999; Zbl 0971.32016)].
Theorem 4.6. (Vitushkin’s Conjecture) Let \(S\subset{\mathbb C}P^2\) be an embedded \(2\)-sphere that is not homologous to zero. Then every function holomorphic in a connected neighborhood of \(S\) in \({\mathbb C}P^2\) is constant [see Zametki 63, No. 4, 599-606 (1998); translation in Math. Notes 63, No. 3-4, 527-532 (1998; Zbl 0933.32045)].
The paper concludes with two open problems: (f) Are all exotic Stein \({\mathbb R}^4\)’s diffeomorphic to domains in \({\mathbb R}^4\), or (g) even any or all of them biholomorphic to domains in \({\mathbb C}^2\)?
The paper is informative and very pleasant to read.
For the entire collection see [Zbl 0972.00032].
32Q28 Stein manifolds
32E10 Stein spaces
32Q55 Topological aspects of complex manifolds