Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A long $\mathbb C^2$ without holomorphic functions (1511.05075v4)

Published 16 Nov 2015 in math.CV

Abstract: In this paper we construct for every integer $n>1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\mathbb Cn$ (i.e., a long $\mathbb Cn$), but it does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new biholomorphic invariants of a complex manifold $X$, the stable core and the strongly stable core, that 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 $B\subset \mathbb Cn$ which is the closure of its interior is the strongly stable core of a long $\mathbb Cn$; in particular, biholomorphically nonequivalent sets give rise to nonequivalent long $\mathbb Cn$'s. Furthermore, for any open set $U\subset \mathbb Cn$ there exists a long $\mathbb Cn$ whose stable core is dense in $U$. It follows that for any $n>1$ there is a continuum of pairwise nonequivalent long $\mathbb Cn$'s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long standing open problems.

Summary

We haven't generated a summary for this paper yet.