Non-Kähler complex structures on $R^4$

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 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.

9 pages, 3 figures, minor revision, Definition 1 has been changed