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Infinitesimal symmetries of weakly pseudoconvex manifolds

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Abstract

We consider weakly pseudoconvex hypersurfaces with polynomial models in \({\mathbb {C}}^N\) and their symmetry algebras. In the most prominent case of special models, given by sums of squares of polynomials, we give their complete classification. In particular, we prove that such manifolds do not admit any nonlinear symmetries, depending only on complex tangential variables, nor do they admit real or nilpotent linear symmetries. This leads to a sharp 2-jet determination result for local automorphisms. We also give partial results in the general case and a more detailed description of the graded components in complex dimension three. The results also provide an important necessary step for solving the local equivalence problem on such manifolds.

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Authors and Affiliations

  1. Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, South Korea

    Shin-Young Kim

  2. Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic

    Martin Kolář

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  1. Shin-Young Kim
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  2. Martin Kolář
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Correspondence to Martin Kolář.

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S.-Y. Kim was supported by the GACR Grant GA17-19437S and by IBS-R003-D1 Institute for Basic Science in Korea. M. Kolář was supported by the GACR Grants GA17-19437S and GA21-09220S.

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Kim, SY., Kolář, M. Infinitesimal symmetries of weakly pseudoconvex manifolds. Math. Z. 300, 2451–2466 (2022). https://doi.org/10.1007/s00209-021-02873-w

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