Conull hypersurfaces in Minkowski space

Abstract

Abstract. A submanifold of M = Gr(2, C4) is conull when its conormal space is in the kernel of the dualized conformai metric of M. We show that there are no conull compact complex 3-dimensional submanifolds of M.Let M be complex Minkowski space, complex analytically homeomorphic to Gr(2, C4), the Grassmannian of 2-planes in C4.Each linear P2 in M is a null manifold i.e. the tangent space TX(P2) consists entirely of null vectors, for each x E P2 cf. Let A be a complex submanifold of M. We say that A is conull if the conormal space of X,N*XC T*M, consists entirely of null covectors. It is an exercise in linear algebra to verify that for dim(A) = 2, Ais null precisely when Ais conull. However, the plethora of null curves is in marked contrast to the lack of conull hypersurfaces.Theorem. Let X be a smooth compact complex hypersurface of M. Then X is not conull.Proof. Let G = PGL(3, C) be the projective linear group. The group G acts transitively on M, and defines an action on the cotangent bundle, T*M. This latter action has 3 orbits-the zero section, the null covectors, and the open orbit. It is well known that the normal bundle, TVA, of A in M is ample (e.g. [3, 2.9]