Uniqueness of complex contact structures

Abstract

Let X be a complex Fano-manifolds with second Betti-number 1 which carries a contact structure. It follows from previous work that such a manifold can always be covered by lines. Thus, it seems natural to consider the geometry of lines in greater detail. In this brief note we show that if x in X is a general point, then all lines through x are smooth. If X is not the projective space, then the tangent spaces to lines generate the contact distribution at x. As a consequence we obtain that the contact structure on X is unique, a result previously obtained by C. LeBrun in the case that X is a twistor space.