Equivalences of Real Submanifolds in Complex Space

Abstract

We show that for any real-analytic submanifold M in ℂ^N there is a proper real-analytic subvariety V ⊂ M such that for any p ∊ M \ V, any realanalytic submanifold M^′ in ℂ^N, and any p^′ ∊ M^′, the germs (M, p) and (M^′, p^′) of the submanifolds M and M^′ at p and p^′ respectively are formally equivalent if and only if they are biholomorphically equivalent. As an application, for p ∊ M \ V, the problem of biholomorphic equivalence of the germs (M, p) and (M^′, p^′) is reduced to that of solving a system of polynomial equations. More general results for k-equivalences are also stated and proved.