Abstract.
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 \({\frak g}\) is a simple Lie algebra, the unique closed orbit in \({\bold P}({\frak g})\) (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(H 0(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.
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Received: July 28, 1997
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Beauville, A. Fano contact manifolds and nilpotent orbits. Comment. Math. Helv. 73, 566–583 (1998). https://doi.org/10.1007/s000140050069
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DOI: https://doi.org/10.1007/s000140050069