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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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References:
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