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2016 A long $\mathbb{C}^2$ without holomorphic functions
Luka Boc Thaler, Franc Forstnerič
Anal. PDE 9(8): 2031-2050 (2016). DOI: 10.2140/apde.2016.9.2031


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.


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Luka Boc Thaler. Franc Forstnerič. "A long $\mathbb{C}^2$ without holomorphic functions." Anal. PDE 9 (8) 2031 - 2050, 2016.


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

zbMATH: 1368.32008
MathSciNet: MR3599525
Digital Object Identifier: 10.2140/apde.2016.9.2031

Primary: 32E10 , 32E30 , 32H02

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

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.9 • No. 8 • 2016
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