Parabolic Stein manifolds. (English) Zbl 1288.32036
Let \(X\) be a connected Stein manifold. In this paper, different notions of parabolicity of \(X\) are discussed. \(X\) is called \(S\)-parabolic if there exists a plurisubharmonic exhaustion function \(\rho\) that is maximal outside a compact subset of \(X\). If in addition \(\rho\) is continuous, then \(X\) is called \(S^*\)-parabolic.
It is clear that \(S^*\)-parabolicity implies \(S\)-parabolicity and \(S\)-parabolic manifolds are parabolic, i.e. all bounded above plurisubharmonic functions on \(X\) are constants. Moreover all that notions coincide for Stein manifolds of dimension one. For higher dimension this problem remains open.
The main result of this article is the characterization of \(S^*\)-parabolic Stein manifolds in terms of the linear topological structure of the space of holomorphic functions \(\mathcal O(X)\) (the first result of this type was proved by A. Aytuna et al. [Math. Ann. 283, No. 2, 193–202 (1989; Zbl 0643.46001)]. Namely, \(X\) is \(S^*\)-parabolic if and only if \(\mathcal O(X)\) is tamely isomorphic to an infinite type power series space
\[ \Lambda_{\infty}(\alpha_m)=\bigg\{x=(x_m)_{m=0}^{\infty}: |x|^{}_k=\sum_{m=0}^{\infty}|x_m|e^{k\alpha_m}<\infty, \;k=1,2,\dots \bigg\}, \]
for some \(\alpha_m\nearrow \infty\).
The authors provide also some nontrivial examples of parabolic manifolds. Let
\[ X=\mathbb C^n\setminus \big\{z\in \mathbb C^n:F(z)=0\big\}, \] where \(F\) is a Weierstrass polynomial in \(\mathbb C^n\). Then \(X\) is \(S^*\)-parabolic, in particular, the complement of the graph of entire holomorphic function is \(S^*\)-parabolic.
A generalization of J.-P. Demailly’s theorem [Mém. Soc. Math. Fr., Nouv. Sér. 19, 124 p. (1985; Zbl 0579.32012)] is proved: assume that there exists a plurisubharmonic exhaustion function \(\rho\) on a Stein manifold \(X\) of dimension \(n\) such that \[ \liminf_{r\to \infty}\frac {\int_{\{\rho<\ln r\}}(dd^c\rho)^n}{(\ln r)^n}=0, \] then \(X\) is parabolic.
Using the previous result, the parabolicity of Sibony-Wong manifolds [N. Sibony and P.-M. Wong, Ann. Pol. Math. 39, 165–174 (1981; Zbl 0476.32005)] is described.
It is clear that \(S^*\)-parabolicity implies \(S\)-parabolicity and \(S\)-parabolic manifolds are parabolic, i.e. all bounded above plurisubharmonic functions on \(X\) are constants. Moreover all that notions coincide for Stein manifolds of dimension one. For higher dimension this problem remains open.
The main result of this article is the characterization of \(S^*\)-parabolic Stein manifolds in terms of the linear topological structure of the space of holomorphic functions \(\mathcal O(X)\) (the first result of this type was proved by A. Aytuna et al. [Math. Ann. 283, No. 2, 193–202 (1989; Zbl 0643.46001)]. Namely, \(X\) is \(S^*\)-parabolic if and only if \(\mathcal O(X)\) is tamely isomorphic to an infinite type power series space
\[ \Lambda_{\infty}(\alpha_m)=\bigg\{x=(x_m)_{m=0}^{\infty}: |x|^{}_k=\sum_{m=0}^{\infty}|x_m|e^{k\alpha_m}<\infty, \;k=1,2,\dots \bigg\}, \]
for some \(\alpha_m\nearrow \infty\).
The authors provide also some nontrivial examples of parabolic manifolds. Let
\[ X=\mathbb C^n\setminus \big\{z\in \mathbb C^n:F(z)=0\big\}, \] where \(F\) is a Weierstrass polynomial in \(\mathbb C^n\). Then \(X\) is \(S^*\)-parabolic, in particular, the complement of the graph of entire holomorphic function is \(S^*\)-parabolic.
A generalization of J.-P. Demailly’s theorem [Mém. Soc. Math. Fr., Nouv. Sér. 19, 124 p. (1985; Zbl 0579.32012)] is proved: assume that there exists a plurisubharmonic exhaustion function \(\rho\) on a Stein manifold \(X\) of dimension \(n\) such that \[ \liminf_{r\to \infty}\frac {\int_{\{\rho<\ln r\}}(dd^c\rho)^n}{(\ln r)^n}=0, \] then \(X\) is parabolic.
Using the previous result, the parabolicity of Sibony-Wong manifolds [N. Sibony and P.-M. Wong, Ann. Pol. Math. 39, 165–174 (1981; Zbl 0476.32005)] is described.
Reviewer: Rafał Czyz (Krakow)
