Complex Contact Manifolds

While the study of complex contact manifolds is almost as old as the modern theory of real contact manifolds, the subject has received much less attention and as many examples are now appearing in the literature, especially twistor spaces over quaternionic Kähler manifolds (e.g., LeBrun [1991], [1995], Moroianu and Semmelmann [1994], Salamon [1982], Ye [1994]), the time is ripe for another look at the subject. As an indication of this interest we note, for example, the following result of Moroianu and Semmelmann [1994] that on a compact spin Kähler manifold M of positive scalar curvature and complex dimension 4l + 3, the following are equivalent: (i) M is a Kähler—Einstein manifold admitting a complex contact structure, (ii) M is the twistor space of a quaternionic Kähler manifold of positive scalar curvature, (iii) M admits Kählerian Killing spinors. LeBrun [1995] proves that a complex contact manifold of positive first Chern class, i.e., a Fano contact manifold, is a twistor space if and only if it admits a Kähler-Einstein metric and conjectures that every Fano contact manifold is a twistor space.