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

Luka Boc Thaler, Franc Forstneric

Published 2015-11-16Version 1

For every integer $n>1$ we construct a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\mathbb C^n$ (i.e., a long $\mathbb C^n$), but it does not admit any nonconstant holomorphic functions. We also introduce new biholomorphic invariants of a complex manifold, the stable core and the strongly stable core, and prove that every compact strongly pseudoconvex polynomially convex domain $B\subset \mathbb C^n$ is the strongly stable core of a long $\mathbb C^n$; in particular, non-equivalent domains give rise to non-equivalent long $\mathbb C^n$'s. Furthermore, we show that for any open set $U\subset \mathbb C^n$ there exists a long $\mathbb C^n$ whose stable core is dense in $U$. Thus, for any $n>1$ there exist uncountably many pairwise non-equivalent long $\mathbb C^n$'s.These results answer several long standing open problems.