Reflection Ideals and mappings between generic submanifolds in complex space

Abstract

Results on finite determination and convergence of formal mappings between smooth generic submanifolds inC N are established in this article. The finite determination result gives suffi- cient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces inC N are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of reflection ideals associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds. 1. Introduction and main results In this article, we study formal mappings between smooth generic submanifolds in C N and establish results on finite determination, convergence and local biholomorphic, and algebraic equivalence. Our finite determination result gives sufficient conditions to guarantee that a formal map as above is uniquely determined by its jet (of a preassigned order) at a point. For real-analytic generic submanifolds, we prove convergence of formal mappings under appropriate assumptions and also give natural geometric conditions to assure that if two germs of such submanifolds are formally equivalent, then they are necessarily biholomorphically equivalent. If the submanifolds are moreover real-algebraic, we address the question of deciding when biholomorphic equivalence implies algebraic equivalence. In particular, we prove that if two real-algebraic hypersurfaces in C N are biholomorphically equivalent, then they are in fact algebraically equivalent. All the results are first proved in the more general context of reflection ideals associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds. We now give precise definitions in order to state some of our main results. Let p 2 C N and