A long ℂ2 without holomorphic functions
| Autor: | Franc Forstneric, Luka Boc Thaler |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Numerical Analysis
Pure mathematics Mathematics::Complex Variables Continuum (topology) Applied Mathematics 010102 general mathematics Fatou–Bieberbach domain Open set Holomorphic function Convex set 01 natural sciences Compact space 0103 physical sciences Stein manifold 010307 mathematical physics 0101 mathematics Complex manifold Analysis Mathematics |
| Zdroj: | Analysis & PDE. 9:2031-2050 |
| ISSN: | 1948-206X 2157-5045 |
| DOI: | 10.2140/apde.2016.9.2031 |
| Popis: | We construct for every integer n > 1 a complex manifold of dimension n which is exhausted by an increasing sequence of biholomorphic images of ℂn (i.e., a long ℂn), 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 ⊂ ℂn such that B = B∘ ¯ is the strongly stable core of a long ℂn; in particular, holomorphically nonequivalent sets give rise to nonequivalent long ℂn’s. Furthermore, for every open set U ⊂ ℂn there exists a long ℂn whose stable core is dense in U. It follows that for any n > 1 there is a continuum of pairwise nonequivalent long ℂn’s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long-standing open problems. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|