Non-Kähler complex structures on $\mathbb{R}^4$

Abstract

We construct the first examples of non-Kähler complex structures on ${ℝ}^4$. These complex surfaces have some analogies with the complex structures constructed in the 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.