Convergence and divergence of formal CR mappings

Abstract

A formal holomorphic map H: (M,p)!M ′ from a germ of a real-analytic submanifold M⊂C at p∈M into a real-analytic subset M ′⊂CN ′ is an N ′-tuple of formal holomorphic power series H=(H1, ...,HN ′) satisfying H(p)∈M ′ with the property that, for any germ of a real-analytic function δ(w, w) at H(p)∈C ′ which vanishes on M ′, the formal power series δ(H(z), H(z)) vanishes on M . There is an abundance of examples showing that formal maps may diverge: After the trivial example of self-maps of a complex submanifold, possibly the simplest non-trivial example is given by the formal maps of (R, 0) into R which are just given by the formal power series in z∈C with real coefficients, that is, by elements of R[[z]]. It is a surprising fact at first that, for formal mappings between real submanifolds in complex spaces, if one assumes that the trivial examples above are excluded in a suitable sense, the situation is fundamentally different. The first result of this kind was encountered by Chern and Moser in [CM], where—as a byproduct of the convergence of their normal form—it follows that every formal holomorphic invertible map between Levinon-degenerate hypersurfaces in C necessarily converges. The convergence problem, that is, deciding whether formal maps, as described above, are in fact convergent, has been studied intensively in different contexts, both for CR manifolds and for manifolds with CR singularities, for which we refer the reader to the papers [Rot], [MMZ2], [LM1], [HY1], [HY2], [HY3], [Sto], [GS] and the references therein. Solutions to the convergence problem have important applications, for example, to the biholomorphic equivalence