A long ℂ2 without holomorphic functions

Boc Thaler, Luka; Forstnerič, Franc

doi:10.2140/apde.2016.9.2031

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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

DOI: 10.2140/apde.2016.9.2031

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 $B = \bar{B^{\circ}}$ 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

Milestones

Received: 4 March 2016

Revised: 19 July 2016

Accepted: 28 August 2016

Published: 11 December 2016

Authors

Luka Boc Thaler
Faculty of Education University of Ljubljana Kardeljeva ploščad 16 SI-1000 Ljubljana Slovenia	Institute of Mathematics, Physics and Mechanics Jadranska 19 SI-1000 Ljubljana Slovenia

Franc Forstnerič
Faculty of Mathematics and Physics University of Ljubljana Jadranska 19 SI-1000 Ljubljana Slovenia	Institute of Mathematics, Physics and Mechanics Jadranska 19 SI-1000 Ljubljana Slovenia

Vol. 9, No. 8, 2016