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

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.