Quasi-complete homogeneous contact manifold associated to a cubic form
  Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic form is the determinant of a simple cubic Jordan algebra. 
Jun-Muk Hwang (KIAS), Laurent Manivel (IF)