Abstract:
In 1907 Poincare formulated his “Problem local”: for given germs of real-analytic hypersurfaces in complex two-space find all local biholomorphic maps between them. This problem has important application to Several compelx Variables, since the study of mappings between domain in complex space can be reduced to that of local maps between their boundaries. Poincare's question naturaly splits to the equivalence problem for two given germs, and to the problem of describing local automorphisms of real-analytic hypersurfaces. Poincare did a substantial progress in solving both problems, by showing first the two germs in general position are inequivalent, and second by showing that the dimension of the symmetry group of a germ in the Levi-nondegenerate case does not exceed 8. More detailed results in the Levi-nondegenerate case were obtained in further work of Cartan, Tanaka, Chern and Moser, and Beloshapka. However, for hypersurfaces with Levi degeneracies the question on possible automorphism groups remained unsolved. In the finite type case (i.e., when a real hypersurface does not contain germs of complex hypersurfaces) the problem was solved independently by Beloshapka, Ejov and Kolar. They showed that the dimension of the group does not exceed 4. However, their method (e.g., the method of "polynomial models") is not applicable to the infinite type case, i.e., when a real hypersurfaces contains a complex germ. It was a long-standing problem to obtained the description in the infinite type case. In our work with Shafikov, we developed a method of solving this problem by using connections between real hypersurfaces and second order complex differential equations. The infinite type case corresponds in this way to ODEs with an isolated singularity. By studying symmetries of an appropriate class of second order singular ODEs, we were able to classify all hypersurfaces with groups of dimension 4 and higher. It turns out that there is a gap for the dimensions which looks as $\dim=\infty,8,5,4,3,2,1,0$ (this gap was conjectured by Beloshapka and known as the Dimension Conjecture).
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