arXiv:math/9901007v1 [math.CV] 4 Jan 1999 On partial analyticity of CR mappings Bernard COUPET, Sergey PINCHUK 1 and Alexandre SUKHOV Abstract. We study the problem of holomorphic extension of a smooth CR mapping from a real analytic hypersurface to a real algebraic set in complex spaces of different dimensions. AMS Mathematics Subject Classification: 32D15,32D99,32F25,32H99 Key words: CR mapping, holomorphic extension, reflection principle, transcendence degree 1 Introduction Let X and Y be real analytic Cauchy - Riemann manifolds in complex affine spaces (of different dimensions, in general), and f : X −→ Y be a smooth CR mapping. It is natural to ask under what conditions f is real analytic (and, therefore, extends holomorphically to a neighborhood of X)? The intrinsic tool to study this problem is the reflection principle. It has two major variations: the analytic one (which makes use of the tangent Cauchy - Riemann operators) and the geometric one (involving a study of analytic geometry of the Segre families). In the present paper we assume that Y is a real algebraic variety and use the analytic approach. Algebraicity of Y allows to involve additionaly methods of commutative algebra. A local character of the problem makes it more convenient to formulate some of our results in terms of germs of CR mappings. Let X be a real analytic hypersurface in I C n , minimal at a point p ∈ X. As usual, the minimality means that X contains no germs of complex hypersurfaces in a neighborhood of p. Let also Y be a real algebraic subset of I C N i.e. the zero set of finite number of real valued polynomials in I C N . Fix a point p ∈ X and consider a germ of a smooth (everywhere below this means C ∞ ) CR mapping p f : X −→ Y of X to Y . This means that there exists a neighborhood U of p in I C n and a representative mapping f of p f defined on X ∩ U such that f (X ∩ U ) ⊂ Y . Consider also the field M p (X) of restrictions to X of germs of meromorphic functions at p and the finite type extension M p (X)( p f 1 ,..., p f N ) of this field generated by the components of the germ p f . Denote by tr.deg.( p f ) the transcendence degree of this extension over M p (X) (see section 2 for definitions). The transcendence degree measures ”the degree of non - analyticity” of p f . If tr.deg. p f is equal to m, we will show that the graph of p f is contained in a complex (n + m) - dimensional variety in I C n+N . In particular, we will show that p f is real analytic if and only if tr.deg. p f = 0. Let f : X −→ I C N be a smooth CR mapping of a real analytic minimal hypersurface. There exists an open dense subset of X where the rank of f achieves its maximal value, which we call the generic rank of f . As usual, by rank p f (the rank of the germ p f ) we mean a generic rank of its representative mapping; this definition does not depend on the choice of a representative. The 1 The second author is partially supported by the NSF grant DMS 96 225 94 1