Using the analytic theory of differential equations, we construct examples of formally but not holomorphically equivalent real-analytic Levi nonflat hypersurfaces in C n together with examples of such hypersurfaces with divergent formal CR-automorphisms.

Abstract. We establish an injective correspondence M −→ E (M ) between real-analytic nonminimal hypersurfaces M ⊂ C 2 , spherical at a generic point, and a class of second order complex ODEs with a meromorphic singularity. We apply this result to the proof of the bound dim hol(M, p) ≤ 5 for the infinitesimal automorphism algebra of an arbitrary germ (M, p) ∼ (S 3 , p ′ ) of a real-analytic Levi nonflat hypersurface M ⊂ C 2 (the Dimension Conjecture). This bound gives the proof of the dimension gap dim hol(M, p) = {8, 5, 4, 3, 2, 1, 0} for the dimension of the automorphism algebra of a real-analytic Levi nonflat hypersurface. As another application we obtain a new regularity condition for CR-mappings of nonminimal hypersurfaces, that we call Fuchsian type, and prove its optimality for extension of CR-mappings to nonminimal points. We also obtain an existence theorem for solutions of a class of singular complex ODEs.

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We study the analytic continuation problem for a germ of a biholomorphic mapping from a nonminimal real hypersurface M ⊂ C n into a real hyperquadric Q ⊂ CP n and prove that under certain nondegeneracy conditions any such germ extends locally biholomorphically along any path lying in the complement U \ X of the complex hypersurface X contained in M for an appropriate neighbourhood U ⊃ X. Using the monodromy representation for the multiple-valued mapping obtained by the analytic continuation we establish a connection between nonminimal real hypersurfaces and singular complex ODEs.

The paper considers a class of Lagrangian surfaces in C 2 with isolated singularities of the unfolded Whitney umbrella type. We prove that generically such a surface is locally polynomially convex near a singular point of this kind.MSC: 32E20, 32E30, 32V40, 53D12.