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A
long $\mathbb{C}^2$ without holomorphic functions
Luka Boc Thaler and Franc Forstnerič |
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Vol. 9 (2016), No. 8, 2031–2050
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Abstract |
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We construct for every integer a complex manifold of dimension which is exhausted by an increasing sequence of biholomorphic images of (i.e., a long ), but does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new holomorphic invariants of a complex manifold , 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 . We show that every compact polynomially convex set such that is the strongly stable core of a long ; in particular, holomorphically nonequivalent sets give rise to nonequivalent long ’s. Furthermore, for every open set there exists a long whose stable core is dense in . It follows that for any there is a continuum of pairwise nonequivalent long ’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
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Mathematical Subject Classification 2010
Primary: 32E10, 32E30, 32H02
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Milestones
Received: 4 March 2016
Revised: 19 July 2016
Accepted: 28 August 2016
Published: 11 December 2016
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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 |
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