4-variétés parallélisables sans structure complexe dont l'espace twistoriel est complexe

The aim of this Note is to give some applications of twistor theory about existence or non-existence of complex structures. We slightly improve Yau's result [Topology 15 (1976) 51-53] by giving the full list of compact parallelizable real 4-manifolds with a complex structure. On the other hand, we give a family of parallelizable 4-manifolds without complex structure but whose product with the sphere S2 is complex. To cite this article: G. Deschamps, C. R. Acad. Sci. Paris, Ser. I 341 (2005).