Serre problem and Inoue-Hirzebruch surfaces (Q5929691)

This paper treats the so-called Serre problem of Stein theory [\textit{J.-P. Serre}, Colloque fonctions plusieurs variables, Brussels, 1953, 57-68 (1953; Zbl 0053.05302)]: if \(E\to B\) is a (locally trivial) holomorphic fiber bundle with fiber type \(F\), where \(B\), \(F\) are Stein manifolds, is \(E\) also a Stein manifold? There are strong positive results on this: \textit{K. Stein} [Arch. Math. 7, 354-361 (1956; Zbl 0072.08002)] proved that \(E\) is Stein if \(F\) is zero-dimensional. \textit{Y. Matsushima} and \textit{A. Morimoto} [Bull. Soc. Math. Fr. 88, 137-155 (1960; Zbl 0094.28104)] showed that \(E\) is Stein if the structure group of \(E\) is a connected complex Lie group. \textit{J.-L. Stehlé} [C. R. Acad. Sci., Paris, Sér. A 279, 235-238 (1974; Zbl 0287.32013); Lect. Notes Math. 474, 155-179 (1975; Zbl 0309.32011)] showed that \(E\) is Stein if \(F\) is so-called hyperconvex. \textit{J. E. Fornæss} and \textit{K. Diederich} [Proc. Natl. Acad. Sci. USA 72, 3279-3280 (1975; Zbl 0309.32012); Invent. Math. 39, 129-141 (1977; Zbl 0353.32025)] showed that \(E\) is Stein if \(F\) is a bounded Stein open subset of some \({\mathbb C}^n\) with \(C^2\)-smooth boundary by proving that such an \(F\) is hyperconvex. \textit{Y.-T. Siu} [Math. Ann. 219, 171-192 (1976; Zbl 0318.32010)] proved that \(E\) is Stein if \(F\) is a bounded Stein open subset of some \({\mathbb C}^n\) and the first Betti number of \(F\) is zero. \textit{N. Mok} [Math. Ann. 258, 145-168 (1981; Zbl 0497.32013)] showed that \(E\) is Stein if \(F\) is one-dimensional. There are, however, counterexamples also. The first one is by \textit{H. Skoda} [C. R. Acad. Sci., Paris, Sér. A 284, 1199-1202 (1977; Zbl 0353.32032); Invent. Math. 43, 97-107 (1977; Zbl 0365.32018)], who gave an \(E\) not Stein with \(B={\mathbb C}\), \(F={\mathbb C}^2\). Skoda's example was modified by \textit{J.-P. Demailly} [Invent. Math. 48, 293-302 (1978; Zbl 0372.32012)], who had for \(B\) a disc or \({\mathbb C}\), \(F={\mathbb C}^2\). In the influential survey article [Bull. Am. Math. Soc. 84, 481-512 (1978; Zbl 0423.32008)], \textit{Y.-T. Siu} conjectured that if in addition \(F\) is a bounded pseudoconvex domain in some \({\mathbb C}^n\), then \(E\) will necessarily be Stein. The counterexample of \textit{G. Cœuré} and \textit{J.-J. Loeb} in [Ann. Math. (2) 122, 329-334 (1985; Zbl 0585.32030)] shows that this is not the case: they produced an \(E\) not Stein with \(B={\mathbb C}\setminus\{0\}\), and \(F\) a pseudoconvex bounded Reinhardt domain in \({\mathbb C}^2\). The goal of the article under review is to illuminate the counterexample of Cœuré and Loeb in the light of geometrical and combinatorial properties of the so-called Inoue-Hirzebruch surfaces. These surfaces relate to cusps and Reinhardt domains in \({\mathbb C}^2\) of the form \(\{(u,v):0<|u|^\alpha<|v|<|u|^\beta\}\) (such as the \(F\) of Cœuré and Loeb). Part 1 introduces the Inoue--Hirzebruch surfaces as surfaces defined in terms of blow-ups governed by a so-called Dloussky-matrix. A Dloussky-matrix is one that can be written as \(A={ 0 1\choose 1 k_1} \ldots { 0 1\choose 1 k_N}\) with integers \(k_i\geq 1\), \(1\leq i\leq N\) -- such a representation of \(A\) is unique. \S 3 defines an involution (called transposition) on the set of all Inoue-Hirzebruch surfaces, and makes use of it to provide a counterexample to `conjecture 3' of \textit{I. Nakamura} [Tôhoku Math. J. 42, 475-516 (1990; Zbl 0732.14019)] that small neighborhoods of the maximal divisor determine the surface. (The author credits G. Dloussky and K. Oeljeklaus for an earlier (unpublished) counterexample to the same conjecture of Nakamura.) Part 2 treats the Serre problem proper. The main result is that if \(A\) is a Dloussky-matrix with \(N\) even, then the construction and the proof of Cœuré and Loeb can be adapted and carried out, and the resulting fiber bundle \(E\to B={\mathbb C}\setminus\{0\}\) is not Stein, and has fiber type \(F\) a pseudoconvex bounded Reinhardt domain (related to Inoue-Hirzebruch surfaces). The technique of proof is an analysis of the form of holomorphic functions on \(E\) in terms of Laurent series. It turns out (Proposition 6.1) that the potential theory method from Cœuré and Loeb implies that certain monomials are absent from the Laurent series of holomorphic functions on \(E\). This fact is then applied to show that \(E\) is not Stein. A consequence of the analysis is the curious fact (Corollary 6.4) that if \(g\) holomorphic on \({\mathbb C}\) is such that \(g(z+j)e^{e^{j^2}}\to 0\) uniformly on compacts as \(j\to\pm\infty\), then \(g\equiv 0\). The last point (Proposition 8.3) is that the manifold \(E\) has no Stein envelope of holomorphy. The paper is very readable and its arguments are mostly geometric and combinatorial in nature.