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

TL;DR

This paper constructs complex manifolds called long a2^na2 that lack nonconstant holomorphic functions, introduces new invariants to classify them, and demonstrates their diverse structures, answering longstanding open problems.

Contribution

It introduces the concept of long a2^na2, constructs examples with unique properties, and develops new invariants to distinguish these manifolds.

Findings

Existence of long a2^na2 without nonconstant holomorphic functions

Introduction of stable core and strongly stable core invariants

Presence of a continuum of nonequivalent long a2^na2 with specific properties

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 $\mathbb C^n$ (i.e., a long $\mathbb C^n$), 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 $B\subset \mathbb C^n$ which is the closure of its interior is the strongly stable core of a long $\mathbb C^n$; in particular, biholomorphically nonequivalent sets give rise to nonequivalent long $\mathbb C^n$'s. Furthermore, for any open set $U\subset \mathbb C^n$ there exists a long $\mathbb C^n$ whose stable core is dense in $U$. It follows that for any $n>1$ there is a continuum of pairwise nonequivalent long $\mathbb C^n$'s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long standing open problems.