Equivalences of real submanifolds in complex space

Abstract

We show that for any real-analytic submanifold M in ℂN there is a proper real-analytic subvariety V ⊂ M such that for any p ∈ M\V, any realanalytic submanifold M′ in ℂN, 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. © Applied Probability Trust 2001.