Fano contact manifolds and nilpotent orbits

  • Ting-Yu Lee

    Ecole Normale Superieure, Paris, France


A contact structure on a complex manifold M is a corank 1 subbundle F of TM such that the bilinear form on F with values in the quotient line bundle L = TM/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{\frak g} is a simple Lie algebra, the unique closed orbit in P(g){\bold P}({\frak 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--\rightarrow P(H0(M, L)*) 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