  A contact structure on a complex manifold M is a corank 1 subbundle F of T(M)
such that the bilinear form on F with values in the quotient line bundle
L=T(M)/F deduced from the Lie bracket of vector fields is everywhere
non-degenerate. In this paper we consider the case where M is a Fano manifold;
this implies that L is ample. If g is a simple Lie algebra, the unique closed
orbit in P(g) (for the adjoint action) is a Fano contact manifold; it is
conjectured that every Fano contact manifold is obtained in this way. A
positive answer would imply an analogous result for compact quaternion-Kahler
manifolds with positive scalar curvature, a longstanding question in Riemannian
geometry. In this paper we solve the conjecture under the additional
assumptions that the group of contact automorphisms of M is reductive, and that
the image of the rational map M - - -> P(H0(M,L)*) associated to L has maximum
dimension. The proof relies on the properties of the nilpotent orbits in a
semi-simple Lie algebra, in particular on the work of R. Brylinski and B.
Kostant.
