Analysis & PDE

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

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

Luka Boc Thaler and Franc Forstnerič

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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.

Article information

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

Dates
Received: 4 March 2016
Revised: 19 July 2016
Accepted: 28 August 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
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

Subjects
Primary: 32E10: Stein spaces, Stein manifolds 32E30: Holomorphic and polynomial approximation, Runge pairs, interpolation 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions

Keywords
holomorphic function Stein manifold long $\mathbb C^n$ Fatou–Bieberbach domain Chern–Moser normal form

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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