2003/06/17 by Stefan Kebekus, Kebekus, Stefan
Mathematics · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #math.AG #math.DG
paper · pdf · doi:10.48550/arxiv.math/0306260
Fixed a number of minor issues found by the referee. To appear in Compositio Math. A PDF-file with additional graphics is available on the internet at http://www.mi.uni-koeln.de/~kebekus/publications-e.html
arxiv created 2003/11/12 · arxiv updated 2009/11/30
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.