Projective Contact Manifolds I

A (projective) manifoldX of dimension 2n+1 together with a subbundle F ⊂ TX of rank 2n is a contact manifold if the pairing F × F → TX/F =: L defined by the Lie bracket is everywhere non-degenerate. An alternative description is as follows: there is a (sub-)line bundle L ⊂ ΩX such that the induced contact form θ ∈ H(X,ΩX⊗L) is non-degenerate in the sense that in local coordinates the form θ ∧ (dθ) has no zeroes. Contact structures first came up in real geometry. One of the interests in complex geometry lies in the connection with twistor spaces and quaternionic Kähler manifolds. We refer to [LeB95] for an excellent introduction to these matters. This paper contributes to the classification of projective contact manifolds. It has been proved by [Dru98] that κ(X) = −∞. Therefore it seems natural to apply Mori theory for the classification. In section 2 we suppose that KX is not nef, i.e. that there is a curve C ⊂ X such that KX .C < 0. Then there is a “contraction of an extremal ray” φ : X → Y , i.e. a is surjective map φ with connected fibers onto a normal projective variety Y , where −KX is φ-ample and the relative Picard number ρ(X/Y ) = 1 (which just means that the Betti number b2 drops by 1). If 0 < dimY < dimX , we show that φ is a linear Pk-bundle. We believe that the case where φ is birational does not occur. We prove this if φ is generically the blow-up along a submanifold in the manifold Y , or, more generally, if some nontrivial fiber of φ has sufficiently many rational curves of minimal degree (e.g. the fiber is a projective space or a quadric, see remark 2.4).