M. S. Baouendi

Algebraicity of holomorphic mappings between real algebraic sets in Cn

0. Introduction 1. Holomorphic nondegenera~y of real-analytic manifolds 1.1. Preliminaries on real submanifolds of C N 1.2. Holomorphic nondegeneracy and its propagation 1.3. The Levi number and essential finiteness 1.4. Holomorphic nondegeneracy of real algebraic sets 2. The Segre sets of a real~analytic CR submanifold 2.1. Complexification of M, involution, and projections 2.2. Definition of the Segre sets of M at p0 2.3. Homogeneous submanifolds of CR dimension 1 2.4. Homogeneous submanifolds of arbitrary CR dimension 2.5. Proof of Theorem 2.2.1 3. Algebraic properties of holomorphic mappings between real algebraic sets 3.1. A generalization of Theorems 1 and 4 3.2. Propagation of algebralcity 3.3. Proof of Theorem 3.1.2 3.4. Proof of Theorem 3.1.8 3.5. An example 3.6. Proofs of Theorems 1 through 4

Local geometric properties of real submanifolds in complex space

We survey some recent results on local geometric properties of real submanifolds of complex space. Our main focus is on the structure and properties of mappings between such submanifolds. We relate these results to the classification of real submanifolds under biholomorphic, algebraic, or formal transformations. Examples and open problems in this context are also mentioned.

Convergence and finite determination of formal CR mappings

In this paper, we study the convergence and finite determination of formal holomorphic mappings of (C , 0) taking one real submanifold into another. By a formal (holomorphic) mapping H : (C , 0) → (C , 0), we mean an N -vector H = (H1, . . . , HN ), where each Hj is a formal power series in N indeterminates with no constant term. If M and M ′ are real smooth submanifolds through 0 in C defined near the origin by ρ(Z, Z) = 0 and ρ′(Z, Z) = 0 respectively, where ρ and ρ′ are vector valued smooth defining functions, then we say that a formal mapping H : (C , 0)→ (C , 0) sends M into M ′ if the vector valued power series ρ′(H(Z), H(Z)) is a (matrix) multiple of ρ(Z, Z). For real-analytic hypersurfaces, we shall prove the following. Theorem 1. Let M and M ′ be real-analytic hypersurfaces through the origin in C , N ≥ 2. Assume that neither M nor M ′ contains a nontrivial holomorphic subvariety through 0. Then any formal mapping H : (C , 0)→ (C , 0) sending M into M ′ is convergent. The condition that M ′ above does not contain a nontrivial holomorphic subvariety is necessary (see Remark 2.3). As a corollary we obtain the following characterization.

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂN are established in this article. The finite determination result gibes sufficient 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 in ℂN 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.

A general reflection principle in C2

Abstract A real analytic hypersurface M through 0 in Cn is said to have the reflection property if any holomorphic mapping defined on one side of M, not totally degenerate at 0, and mapping M into another real analytic hypersurface in Cn, extends holomorphically to a full neighborhood of 0 in Cn. The main result of this paper is that a real analytic hypersurface in C2 has the reflection property if and only if it is not Levi flat.

Dynamics of the Segre varieties of a real submanifold in complex space

Let M ⊂ C be a smooth (C∞) real submanifold of codimension d with 0 ∈ M . We choose smooth real-valued functions r = (r1, . . . , rd), with differentials dr1, . . . , drd linearly independent at 0, so thatM is defined by r = 0 near the origin. If the complex differentials ∂r1, . . . , ∂rd are also linearly independent at 0, thenM is called generic (near the origin). If M is a real-analytic, generic submanifold, there is a family of complex submanifolds of C , called the Segre varieties associated to M , which carry a great deal of information about the local geometry of M . The Segre varieties have been used by many mathematicians to study mappings between generic submanifolds. (See the end of this introduction for some specific references.) In this paper, we shall consider an algebraic substitute for these varieties for the case of smooth manifolds by introducing a formal mapping which, in the realanalytic case, parametrizes the Segre varieties. Our main objective is to study iterations of this mapping and relate these iterations to the local CR geometry of the manifold. If r is a local defining function of a smooth generic submanifold M as above, then we denote by ρj(Z, Z) the Taylor series of rj at 0. We write ρ = (ρ1, . . . , ρd). We consider ρj(Z, ζ) as a formal power series in the 2N indeterminates (Z, ζ). We shall denote the ring of such power series with complex coefficients by C[[Z, ζ]]. By a formal mapping F : (C, 0) → (C, 0), we shall mean a p-tuple F (x) = (F1(x), . . . , Fp(x)), where x = (x1, . . . , xk), of formal power series Fj ∈ C[[x]] without constant terms. The rank of F , RkF , is defined as the rank of the Jacobian matrix ∂F/∂x regarded as a Kx-linear mapping Kx → Kx, where Kx denotes the field of fractions of C[[x]]. Hence RkF is the largest integer s such that there is an s× s minor of the matrix ∂F/∂x which is not 0 as a formal power series in x. Let γ(ζ, t), where ζ = (ζ1, . . . ζN), t = (t1, . . . , tn), and n = N − d, be a formal mapping (C × C, 0)→ (C , 0) such that

Remarks on the generic rank of a CR mapping

We study germs of smooth CR mappings between embedded real hypersurfaces in complex spaces of the same dimension. In particular, we are interested in the generic rank of such mappings. IfH:M →M′ is a CR map between two hypersurfacesM andM′, we prove that ifM′ does not contain any germ of a holomorphic manifold then eitherH is constant or the generic rank ofH is odd. We also prove that if there is no formal holomorphic vector field tangent toM, then eitherH is constant or genericallyH is a local diffeomorphism. It follows, as a special case, that ifM andM′ are of D-finite type (in the sense of D’Angelo) thenH is either constant or is generically a local diffeomorphism.