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 $\mathfrak{g}$ is a simple Lie algebra, the unique closed orbit in $\mathbf{P}(\mathfrak{g})$ (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 $M--\to \mathbf{P}(H^0(M,L{)}^{\ast })$ sociated 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.