Convergence and divergence of formal CR mappings
The convergence problem of formal holomorphic maps between real analytic submanifolds of complex spaces is studied intensively during the recent years. In particular, the specific case of formal CR maps, i.e., when the source manifold is CR, has received considerable attention. This paper focuses on this case. Among several interesting and significant results, the authors show as the main result that a formal holomorphic map \(H:(M, p)\rightarrow M'\), where \(M\subset\mathbb C^N\) is a generic real analytic submanifold of finite type, \(M'\subset\mathbb C^{N'}\) is a real analytic set and \(N, N'\geq 2\), is convergent whenever \(H(M)\) is not a subset of \(\mathcal E_{M'}\), the collection of all points in \(M'\) which are of infinite D'Angelo type (Theorem 1.1). This significant result is actually a consequence of another one (Theorem 2.2), a powerful result, which provides some general necessary condition for the existence of divergent maps. Accordingly, this theorem states that in order to have a divergent formal map \(H:(M, p)\rightarrow M'\), there must be some integer \(1\leqslant r\leqslant N'\) and, for every positive integer \(k\), a family of \(r\)-dimensional complex analytic subvarieties \((\Upsilon_z)_{z\in M_k}\), parameterized by some neighborhood \(M_k\) of \(p\) in \(M\), such that each submanifold \(\Upsilon_z\) passes throughout \(H(z)\) and has order of contact at least \(k\) with \(M'\) at \(H(z)\). The approach developed in this paper is completely new and different from those used so far to study the convergence problem. The main proofs are based upon establishing a link between divergence properties of formal CR maps \(H:(M, p)\rightarrow M'\) and the geometry of the triple \((M, M', H)\). This is done by means of the so-called \textit{\(k\)-approximate formal deformations}. In terms of this new language, the above mentioned Theorem 2.2 says that, for every \(k\), the existence of \(k\)-approximate formal deformations is a necessary condition for the existence of divergent formal holomorphic maps. Several other interesting results concerning the convergence problem can be found throughout this paper.
scientific article; zbMATH DE number 6925268