FANO MANIFOLDS, CONTACT STRUCTURES, AND QUATERNIONIC GEOMETRY

Source abstract

Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M 4n , g). If Z also admits a second complex contact structure [Formula: see text], then Z=CP 2n+1 . As an application, we give several new characterizations of the Riemannian manifold HP n = Sp(n+1)/(Sp(n)×Sp(1)).