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

TL;DR

This paper introduces the first known examples of non-K"ahler complex structures on four-dimensional Euclidean space, expanding the understanding of complex geometry beyond K"ahler manifolds.

Contribution

It provides a novel construction of non-K"ahler complex structures on R^4, differing from classical examples by Calabi and Eckmann, and broadens the landscape of complex structures in differential geometry.

Findings

First examples of non-K"ahler complex structures on R^4

Construction method differs from classical sphere product examples

Enhances understanding of complex structures in four-dimensional geometry

Abstract

We construct the first examples of non-K\"ahler complex structures on $R^4$. These complex surfaces have some analogies with the complex structures constructed in early Fifties by Calabi and Eckmann on the products of two odd-dimensional spheres. However, our construction is quite different from that of Calabi and Eckmann.