Lines on contact Manifolds IIb

Details Lines on contact Manifolds IIb Lines on contact Manifolds IIb

2003-06-17

arxiv.org/abs/math/0306260v2

Mathematics

Algebraic Geometry

Fixed a number of minor issues found by the referee. To appear in Compositio Math. A PDF-file with additional graphics is avai

Scientific paper

Let X be a complex-projective contact manifold whose second Betti-number is one. It has long been conjectured that X should then be rational-homogeneous, or equivalently, that there exists an embedding of X into a projective space whose image contains lines. Using methods introduced in math.AG/0206193, we show that X is covered by a compact family of rational curves, called "contact lines" that behave very much like the lines on the rational homogeneous examples: if x in X is a general point, then all contact lines through x are smooth, no two of them share a common tangent direction at x, and the union of all contact lines through x forms a cone over an irreducible, smooth base. As a corollary, we obtain that the tangent bundle of X is stable.

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