Fano contact manifolds and nilpotent orbits

  • Ting-Yu Lee

    Ecole Normale Superieure, Paris, France


A contact structure on a complex manifold is a corank subbundle of such that the bilinear form on with values in the quotient line bundle deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where is a Fano manifold; this implies that L is ample.

If is a simple Lie algebra, the unique closed orbit in (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 is reductive, and that the image of the rational map 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.

Cite this article

Ting-Yu Lee, Fano contact manifolds and nilpotent orbits. Comment. Math. Helv. 73 (1998), no. 4, pp. 566–583

DOI 10.1007/S000140050069