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Noncritical holomorphic functions on Stein manifolds. (English) Zbl 1064.32021
This paper on the geometry and topology of Stein manifolds by one of the foremost experts on several complex variables and its geometric aspects is one of the best articles on geometric function theory in recent years, and crowns the author’s many years of work on the homotopy principle for complex submanifolds of Euclidean spaces (so-called Stein manifolds).
According to the paper the author’s main motivation is to generalize to Stein manifolds of higher dimension the following classical theorem of R. C. Gunning and R. Narasimhan [Math. Ann. 174, 103–108 (1967; Zbl 0179.11402)], and to address the long open question whether a parallelizable Stein manifold of dimension \(n>1\) holomorphically immerses in \({\mathbb C}^n\). The result of Gunning and Narasimhan is that on an open Riemann surface (i.e., on a one-dimensional Stein manifold) there is a global holomorphic numerical function with nowhere zero differential, i.e., a noncritical global holomorphic function. Such a function immerses the Riemann surface in \({\mathbb C}\), and is also a submersion.
The cases of submersions and immersions in higher dimensions have different natures. Immersions with positive codimension also have different character than in zero codimension (which could be directly useful in connection with the above mentioned open question). The author in this monumental paper studies the case of holomorphic submersions based in part on the homotopy principle that he himself has developed in recent years after the pattern of the homotopy principle of Gromov and Eliashberg, and in part on many novel approaches introduced here.
The main theorems in the Introduction are the following.
Theorem I. Every Stein manifold admits a holomorphic function without critical points. More precisely, an \(n\)-dimensional Stein manifold admits \([\frac12(n+1)]\) holomorphic functions with pointwise independent differentials, and this number is maximal for certain \(n\)-dimensional Stein manifolds for all \(n\).
Theorem II. (The homotopy principle for holomorphic submersions.) If \(X\) is a Stein manifold of dimension \(n\), \(TX\) its holomorphic tangent bundle, and \(1\leq q<n\), then every continuous surjective map \(TX\to X\times{\mathbb C}^q\) of complex vector bundles is homotopic through maps of the same type to one that is the differential of a holomorphic submersion.
The author’s main results fill in the paper § 2, which runs to nearly \(13\) pages and lists \(12\) theorems that are grouped and can be discussed as follows.
(1) Functions with prescribed critical locus. – Here the main result is that the critical locus can be arbitrarily prescribed including multiplicities, and extension and approximation hold simultaneously, e.g., a noncritical function on a closed complex submanifold can be extended to a global noncritical function. Noncritical functions can serve as local coordinates near any given point, and a noncritical function can separate two given points.
(2) Foliations by complex hypersurfaces. – Here the level sets of noncritical functions are put to use.
(3) Holomorphic submersions and foliations. – Here the first theorem says, in particular, that on a Stein manifold a continuous global coframe of positive corank can be deformed through coframes of the same type to a coframe that is the differential of a holomorphic submersion. Moreover extension and approximation hold simultaneously for such homotopies of coframes.
(4) Existence of homotopies to integrable subbundles. – Here it is shown, in particular, that on a Stein manifold every trivial subbundle \(E\) of positive rank of the tangent bundle that therein has a trivial direct complement is homotopic to an integrable holomorphic subbundle Ker\(\,df\), where \(f\) is a holomorphic submersion to complex Euclidean space. If \(E\) is holomorphic, then the homotopy may be chosen through holomorphic subbundles.
(5) An example. – Here the author presents and discusses an example of Forster that shows that some of the author’s results are sharp. The example is the complement in the projective plane \({\mathbb CP}^2\) of the null cone \(x^2+y^2+z^2=0\) (and its products with itself and possibly with one copy of \({\mathbb C}\)). What is special about this example is that it has the homotopy type of the real projective plane, and thus its characteristic cohomology classes such as its Chern classes are easily computed, and they yield obstructions having certain ranks of trivial subbundles in the tangent or cotangent bundles.
(6) Remarks on parallelizable Stein manifolds. – Here the author gathers some references and makes some comments on parallelizable Stein manifolds, e.g., he poses in particular the question whether every complex hypersurface of \({\mathbb C}^{n+1}\) holomorphically immerses in \({\mathbb C}^n\), as they are parallelizable by an earlier result of the author.
(7) Comparison with smooth immersions and submersions. – In the smooth setting a powerful theory was already in place developed mainly by Smale and Hirsch for immersions, and by Phillips and Gromov for submersions. The author builds on the convex integration lemma of Gromov.
(8) Outline of proofs of the main theorems. – Here the author describes his new approach and powerful new techniques. Some of the proofs are very sophisticated versions of the classical exhaust, patch and approximate methods of Stein theory. The author proves a number of approximation and extension theorems in the course of which one useful main ingredient is the Andersén and Lempert theory of holomorphic automorphisms of \({\mathbb C}^n\), \(n\geq2\), as further developed by Rosay and the author in several papers. One application of this is to push critical points outside a compact set away from the compact set.
The patching/splitting idea pertains to biholomorphisms that are near to the identity, and is with respect to composition over a Cartan pair of compact sets.
The author also utilizes the convex integration lemma of Gromov.
In the exhaustion phases the author makes use of the classical bumping method of Grauert as presented, e.g., by Henkin and Leiterer in their proof of the Oka principle of Grauert without induction on the dimension of the ground manifold. (This proof had an earlier application to the homotopy principle in a paper of Gromov.) This involves the analysis of pasting on handles as one moves past a critical point of a Morse plurisubharmonic exhaustion function.
This paper is clearly a masterpiece and its careful study should be rewarding to the reader.
It may be remarked that the reviewer had the privilege to attend the author’s lecture on the topics of this paper at the Gunning and Kohn conference in 2002.
The language and presentation of the article are excellent and it makes pleasant reading. It also contains some great jokes, e.g., “For the general theory of foliations we refer to [God].” – on page 147 right above Cor. 2.3

MSC:
32Q28 Stein manifolds
32Q55 Topological aspects of complex manifolds
32L05 Holomorphic bundles and generalizations
32A10 Holomorphic functions of several complex variables
32E10 Stein spaces, Stein manifolds
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