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Lines on contact manifolds. (English) Zbl 0983.53031
Complex manifolds \(X\) which carry a complex contact structure, i.e., a non-degenerate subbundle \(F\subset T_X\) of the tangent bundle of corank one, appear naturally as twistor spaces over Riemannian manifolds with quaternionic-Kählerian holonomy group. These manifolds have recently gained considerable interest. The reader is referred to A. Beauvilles’s excellent survey [“Riemannian holonomy and algebraic geometry”, preprint (1999)] for a thorough introduction. Previous results of J. P. Demailly [“On the Frobenius integrability of certain holomorphic \(p\)-forms”, preprint (2000)] and T. Peternell, A. Sommese, J. A. Wisniewski and the author [Invent. Math. 142, 1-15 (2000; Zbl 0994.53024)] practically reduce the study to the case where the contact manifold \(X\) is Fano and where the second Betti-number \(b_2(X)\) is one. Since these manifolds can always be covered by lines, i.e., by rational curves which intersect the ample generator of the Picard-group with multiplicity one, the present paper considers the geometry of these lines in greater detail.
It is shown that if \(x\) is a general point on a contact manifold \(X\), then all lines through \(x\) are smooth. Furthermore, if \(X\) is not the projective space, then the tangent spaces to lines generate the contact distribution \(F\) at \(x\). As a consequence we obtain that the contact structure on \(X\) is unique, a result which was previously conjectured by C. LeBrun.

MSC:
53C28 Twistor methods in differential geometry
53D10 Contact manifolds (general theory)
53C29 Issues of holonomy in differential geometry
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