Fano contact manifolds and nilpotent orbits

Abstract

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.