We investigate holomorphic
Segre-preserving maps sending the complexification ℳ of a generic real analytic
submanifold M ⊆ ℂN of finite type at some point p into the complexification ℳ′ of
a generic real analytic submanifold M′⊆ ℂN′ that is finitely nondegenerate at
some point p′. We prove that for a fixed M and M′, the germs at (p,p) of
Segre submersive holomorphic Segre-preserving maps sending (ℳ,(p,p)) into
(ℳ′,(p′,p′)) rationally depend upon their K-jets at (p,p), for some fixed K
depending only on M and M′, and these maps are uniquely determined by their
K-jets. If, in addition, M and M′ are algebraic, we prove that any such map
must be algebraic. It follows that the set of germs at (p,p) of holomorphic
Segre-preserving automorphisms of the complexification ℳ of a generic
real analytic submanifold M that is finitely nondegenerate and of finite
type at p is an algebraic complex Lie group. We explore the relationship
between this automorphism group and the group of automorphisms of M at
p.
Keywords
holomorphic Segre-preserving map, complexification, real
analytic submanifold, finite jet determination, algebraic
submanifold, finitely nondegenerate, finite type
