Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Abstract

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-K\"ahler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces.