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 homotopy equivalent to a complex torus is biholomorphic to a complex torus.
The question whether a compact complex manifold 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 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.
Citation
Fabrizio Catanese. Keiji Oguiso. Thomas Peternell. "On volume-preserving complex structures on real tori." Kyoto J. Math. 50 (4) 753 - 775, Winter 2010. https://doi.org/10.1215/0023608X-2010-013
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