Defining equations for real analytic real hypersurfaces in 𝐶ⁿ

D’Angelo, John P.

doi:10.1090/S0002-9947-1986-0831189-5

Defining equations for real analytic real hypersurfaces in $\textbf {C}^ n$
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Trans. Amer. Math. Soc. 295 (1986), 71-84 Request permission

Abstract:

A defining function for a real analytic real hypersurface can be uniquely written as $2\operatorname {Re} (H) + E$, where $H$ is holomorphic and $E$ contains no pure terms. We study how $H$ and $E$ change when we perform a local biholomorphic change of coordinates, or multiply by a unit. One of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of $E$ (expanded in terms of $H$) beyond ${E_{00}}$ that serve as obstructions to writing a defining equation as $2\operatorname {Re} (h) + e$, where $e$ is independent of $h$. We also find necessary pluriharmonic obstructions to doing this, which arise from the easier case of attempting to straighten the hypersurface.

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