Vol. 9, No. 8, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 n (i.e., a long 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 n such that B = B ¯ is the strongly stable core of a long n; in particular, holomorphically nonequivalent sets give rise to nonequivalent long n’s. Furthermore, for every open set U n there exists a long n whose stable core is dense in U. It follows that for any n > 1 there is a continuum of pairwise nonequivalent long 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