Complex Contact Manifolds
Stefan Kebekus (Köln)
Motivated by questions coming from Riemannian geometry, complex
contact manifolds have received considerable attention during the last years.
The link between complex and Riemannian geometry is given by the twistor
space construction: twistor spaces over Riemannian manifolds with
quaternion-Kähler holonomy group are complex contact manifolds. As twistor
spaces are covered by rational curves, much of the research is centered about
the geometry of rational curves on the contact spaces.
If X is a complex-projective contact manifold with b_2 = 1, it has
long been conjectured that X should be rational-homogeneous. 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. Among other results, that allows to give a full classification of
contact manifolds with b_2 > 1.