Parametrization of Holomorphic Segre Preserving Maps

In this paper, we explore holomorphic Segre preserving maps. First, we investigate holomorphic Segre preserving maps sending the complexificationM of a generic real analytic submanifold M ⊆ C of finite type at some point p into the complexification M′ of a generic real analytic submanifold M ′ ⊆ C ′ , finitely nondegenerate at some point p. We prove that for a fixed M and M , the germs at (p, p̄) of Segre submersive holomorphic Segre preserving maps sending (M, (p, p̄)) into (M′, (p, p̄)) can be parametrized by their r-jets at (p, p̄), for some fixed r depending only on M and M . (If, in addition, M and M ′ are both real algebraic, then we prove that any such map must be holomorphic algebraic.) From this parametrization, it follows that the set of germs of holomorphic Segre preserving automorphisms H of the complexification M of a real analytic submanifold finitely nondegenerate and of finite type at some point p, and such that H fixes (p, p̄), is an algebraic complex Lie group. We then explore the relationship between this automorphism group and the group of automorphisms of M at p.