Summary: We address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. We prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from geometry of an arbitrary contact manifold. Minimal rational curves on contact manifolds (or contact lines) and their chains are the essential ingredients for our constructions.

14M17, 53C26, 14M20, 14J45

complex contact manifold, Fano variety, minimal rational curves, adjoint variety, Killing form, Lie bracket, Lie algebra grading