On volume-preserving complex structures on real tori

Abstract

A basic problem in the classification theory of compact complex manifolds is to give simple characterizations of complex tori. It is well known that a compact Kähler manifold $X$ homotopy equivalent to a complex torus is biholomorphic to a complex torus.

The question whether a compact complex manifold $X$ diffeomorphic to a complex torus is biholomorphic to a complex torus has a negative answer due to a construction by Blanchard and Sommese.

Their examples, however, have negative Kodaira dimension; thus it makes sense to ask whether a compact complex manifold $X$ with trivial canonical bundle which is homotopy equivalent to a complex torus is biholomorphic to a complex torus.

In this article we show that the answer is positive for complex threefolds satisfying some additional condition, such as the existence of a nonconstant meromorphic function.