Non-K\"ahler complex structures on $R^4$ II

TL;DR

This paper investigates non-Kähler complex structures on R^4, showing they cannot be compactified smoothly, describing their meromorphic functions, Picard groups, and canonical bundles, and demonstrating their existence on all connected non-compact 4-manifolds.

Contribution

It provides a detailed analysis of non-Kähler complex structures on R^4, including their properties and existence on all connected non-compact 4-manifolds, expanding understanding of complex structures in four dimensions.

Findings

Non-Kähler complex surfaces on R^4 cannot be smoothly compactified.

Explicit descriptions of meromorphic functions and canonical bundles are provided.

Picard groups of these surfaces are uncountably infinite.

Abstract

We follow our study of non-K\"ahler complex structures on $R^4$ that we defined in a previous paper. We prove that these complex surfaces do not admit any smooth complex compactification. Moreover, we give an explicit description of their meromorphic functions. We also prove that the Picard groups of these complex surfaces are uncountable, and give an explicit description of the canonical bundle. Finally, we show that any connected non-compact oriented 4-manifold admits complex structures without K\"ahler metrics.