Real hypersurfaces in complex manifolds

TLDR: In this paper, Cartan gave a complete solution of the equivalence problem, which is, among other results, the problem of finding a complete system of analytic invariants for two real analytic real hypersurfaees in Cn+l to be locally equivalent under biholomorphic transformations.

Abstract: Whether one studies the geometry or analysis in the complex number space C a + l , or more generally, in a complex manifold, one will have to deal with domains. Their boundaries are real hypersurfaces of real codimension one. In 1907, Poincar4 showed by, a heuristic argument tha t a real hypersurface in (38 has local invariants unde r biholomorphie transformations [6]. He also recognized the importance of the special uni tary group which acts on the real hyperquadrics (cf. w Following a remark by B. ~Segre, Elie :Cartan took, up again the problem. In t w o profound papers [1], he gave, among other results, a complete solution of the equivalence problem, tha t is, the problem of finding a complete system of analytic invariants for two real analytic real hypersurfaees in C~ to be locally equivalent under biholomorphic transformations. Let z 1, ..., z n+l be the coordinates in Cn+r We s tudy a real hypersurface M at the origin 0 defined by the equation