On volume preserving complex structures on real tori

Details On volume preserving complex structures on real tori On volume preserving complex structures on real tori

2009-12-29

arxiv.org/abs/0912.5313v1

Mathematics

Complex Variables

20 pages, preliminary version of an article to be submitted to a memorial issue of the Journal of Mathematics of Kyoto Univers

Scientific paper

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 K\"ahler manifold $X$ homotopically equivalent to a 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 have however negative Kodaira dimension, thus it makes sense to ask the question whether a compact complex manifold $X$ with trivial canonical bundle which is homotopically equivalent to a complex torus is biholomorphic to a complex torus. In this paper we show that the answer is positive for complex threefolds satisfying some additional condition, such as the existence of a non constant meromorphic function.

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