#### Vol. 9, No. 8, 2016

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A long $\mathbb{C}^2$ without holomorphic functions

### Luka Boc Thaler and Franc Forstnerič

Vol. 9 (2016), No. 8, 2031–2050
##### Abstract

We construct for every integer $n>1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of ${ℂ}^{n}$ (i.e., a long ${ℂ}^{n}$), but does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new holomorphic invariants of a complex manifold $X$, the stable core and the strongly stable core, which are based on the long-term behavior of hulls of compact sets with respect to an exhaustion of $X$. We show that every compact polynomially convex set $B\subset {ℂ}^{n}$ such that $B=\overline{{B}^{\circ }}$ is the strongly stable core of a long ${ℂ}^{n}$; in particular, holomorphically nonequivalent sets give rise to nonequivalent long ${ℂ}^{n}$’s. Furthermore, for every open set $U\subset {ℂ}^{n}$ there exists a long ${ℂ}^{n}$ whose stable core is dense in $U$. It follows that for any $n>1$ there is a continuum of pairwise nonequivalent long ${ℂ}^{n}$’s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long-standing open problems.

##### Keywords
holomorphic function, Stein manifold, long $\mathbb C^n$, Fatou–Bieberbach domain, Chern–Moser normal form
##### Mathematical Subject Classification 2010
Primary: 32E10, 32E30, 32H02