Geodesic
Modification of Poincar\' e's
Izvestiya. Mathematics , Tome 86 (2022) no. 5, pp. 852-875

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The modified Poincaré construction (a generalization of Poincaré's homological operator) was earlier used to estimate the dimension of the local automorphism group for an arbitrary germ of a real-analytic hypersurface in $\mathbf{C}^3$. In the present paper we prove the following alternative. For every hypersurface in $\mathbf{C}^4$, this dimension is either infinite or does not exceed $24$. Moreover, $24$ occurs only for a non-degenerate hyperquadric (one of the two). If the hypersurface is $2$-nondegenerate (resp. $3$-non-degenerate) at a generic point, the bound can be improved to $17$ (resp. $20$).
Keywords: $CR$-manifold, automorphisms, model surfaces. 
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     author = {V. K. Beloshapka},
     title = {Modification of {Poincar\'} e's},
     journal = {Izvestiya. Mathematics },
     pages = {852--875},
     publisher = {mathdoc},
     volume = {86},
     number = {5},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2022_86_5_a2/}
}
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V. K. Beloshapka. Modification of Poincar\' e's. Izvestiya. Mathematics , Tome 86 (2022) no. 5, pp. 852-875. http://geodesic.mathdoc.fr/item/IM2_2022_86_5_a2/