IUMJ

Title: Normal forms and symmetries of real hypersurfaces of finite type in $\mathbb C^2$

Authors: Vladimir Ezhov, Martin Kolar and Gerd Schmalz

Issue: Volume 62 (2013), Issue 1, 1-32

Abstract:

We give a complete description of normal forms for real hypersurfaces of finite type in $\mathbb{C}^2$ with respect to their holomorphic symmetry algebras. The normal forms include refined versions of the constructions by Chern-Moser [S.\:S. Chern and J.\:K. Moser, \textit{Real hypersurfaces in complex manifolds}, Acta Math. \textbf{133} (1974), no. 1, 219--271], Stanton [N.\:K. Stanton, \textit{A normal form for rigid hypersurfaces in $\mathbb{C}^2$}, Amer. J. Math. \textbf{113} (1991), no. 5, 877--910], Kol\'a\v r [M. Kol\'a\v r, \textit{Normal forms for hypersurfaces of finite type in $\mathbb{C}^2$}, Math. Res. Lett. \textbf{12} (2005), no. 5--6, 897--910]. We use the method of simultaneous normalisation of the equations and symmetries that goes back to Lie and Cartan. Our approach leads to a unique canonical equation of the hypersurface for every type of its symmetry algebra. Moreover, even in the Levi-degenerate case, our construction implies convergence of the transformation to the normal form if the dimension of the symmetry algebra is at least two. We illustrate our results by explicitly normalising Cartan's homogeneous hypersurfaces and their automorphisms.