Uniqueness of complex contact structures

Details Uniqueness of complex contact structures Uniqueness of complex contact structures

2000-04-16

arxiv.org/abs/math/0004103v3

Mathematics

Algebraic Geometry

reason for resubmission: improved exposition

Scientific paper

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.

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