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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

Citations:

Zbl 0994.53024
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References:

[1] A. Beauville, Riemannian holonomy and algebraic geometry, LANL-Preprint math.AG/9902110, 1999.
[2] H. Clemens, J. KollaAr and S. Mori, Higher Dimensional Complex Geometry, AsteArisque 16 (1988).
[3] J. P. Demailly, On the Frobenius integrability of certain holomorphic p-forms, LANL-Preprint math.AG/ 0004067, PreApubl. Inst. Fourier 502 (2000).
[4] J. Fogarty and D. Mumford, Geometric Invariant Theory, 2nd edition, Springer Erg. Math. Grenzgeb. 34 (1982). · Zbl 0504.14008
[5] R. Hartshorne, Algebraic Geometry, Springer Grad. Texts Math. 52 (1977).
[6] Hwang J.-M., Math. 486 pp 153– (1997)
[7] Y. Kachi and E. Sato, Polarized varieties whose points are joined by rational curves of small degrees, Ill. J. Math. 43(2) (1999), 350-390. · Zbl 0939.14008
[8] Kebekus S., Invent. Math. 142 pp 1– (2000)
[9] J. KollaAr, Rational Curves on Algebraic Varieties, Springer Erg. Math. Grenzgeb. 3. Folge, 32 (1996).
[10] KollaAr J., Geom. 36 pp 765– (1992)
[11] LeBrun C., Int. J. Math. 3 pp 419– (1995)
[12] C. LeBrun, Twistors for tourists: A pocket guide for algebraic geometers, Kollar, J. et al. (ed.), Algebraic geometry, Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, 1995, Proc. Symp. Pure Math. 62(2) (1997), 361-385. · Zbl 0904.32024
[13] Mori S., Ann. Math. 110 pp 593– (1979)
[14] F. Warner, Foundations of Di erentiable Manifolds and Lie Groups, Scott, Foresman and Company, Glenview, Illinois and London 1971.
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