Complex vector fields and a vanishing theorem for the Euler characteristic

In this paper, we give a more elegant and systematic proof of Jacobowitz’s theorem which relates complex vector fields and the topology of the underlying manifold. The method we use is due to Atiyah [ <italic>Vector fields on manifolds</italic> (English, with French and German summaries), Westdeutscher Verlag, Cologne, 1970], who applied it to the study of real vector fields and characteristic numbers.