## Analysis & PDE

• Anal. PDE
• Volume 9, Number 8 (2016), 2031-2050.

### 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 $ℂn$ (i.e., a long $ℂn$), 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 $B ⊂ ℂn$ such that $B = B∘ ¯$ is the strongly stable core of a long $ℂn$; in particular, holomorphically nonequivalent sets give rise to nonequivalent long $ℂn$’s. Furthermore, for every open set $U ⊂ ℂn$ there exists a long $ℂn$ whose stable core is dense in $U$. It follows that for any $n > 1$ there is a continuum of pairwise nonequivalent long $ℂn$’s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long-standing open problems.

#### Article information

Source
Anal. PDE, Volume 9, Number 8 (2016), 2031-2050.

Dates
Revised: 19 July 2016
Accepted: 28 August 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843382

Digital Object Identifier
doi:10.2140/apde.2016.9.2031

Mathematical Reviews number (MathSciNet)
MR3599525

Zentralblatt MATH identifier
1368.32008

#### Citation

Boc Thaler, Luka; Forstnerič, Franc. A long $\mathbb{C}^2$ without holomorphic functions. Anal. PDE 9 (2016), no. 8, 2031--2050. doi:10.2140/apde.2016.9.2031. https://projecteuclid.org/euclid.apde/1510843382

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