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A few minor clarifications/style fixes
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jirilebl committed Jun 3, 2024
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Showing 1 changed file with 8 additions and 7 deletions.
15 changes: 8 additions & 7 deletions scv.tex
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@@ -9541,14 +9541,15 @@ \section{CR functions}

Real-analytic CR functions on a real-analytic
hypersurface $M$ always extend to holomorphic functions of a neighborhood of $M$.
To prove this we wish to complexify everything, that is treat the
To prove this fact, we complexify everything, that is, we treat the
$z$s and $\bar{z}$s as separate variables. The standard way of
writing a hypersurface as a graph is not as convenient for this setting, so
let us prove that for a real-analytic hypersurface, we can write it as a
graph of a holomorphic function in the complexified variables. That is,
we prove that a real-analytic hypersurface is a
graph of a holomorphic function in the complexified variables restricted to
the diagonal. That is,
using variables $(z,w)$,
we will write $M$ as a graph of $\bar{w}$ over $z$, $\bar{z}$, and $w$.
This allows us to eliminate $\bar{w}$ in any real-analytic
we write $M$ as a graph of $\bar{w}$ over $z$, $\bar{z}$, and $w$.
We can then eliminate $\bar{w}$ in any real-analytic
expression.

\begin{prop} \label{prop:complexificationofrasurface}
@@ -9595,7 +9596,7 @@ \section{CR functions}
$f(z,\zeta,w,\omega)$ vanishes on $\sM$ near the origin.
\end{prop}

Again as a slight abuse of notation $\Phi$ refers to both the function
Again as a slight abuse of notation, $\Phi$ refers to both the function
$\Phi(z,\zeta,w)$ and $\Phi(z,\bar{z},w)$.

\begin{proof}
@@ -9785,7 +9786,7 @@ \section{CR functions}
defined on a neighborhood of the circle and
equal to $f$ on the circle.
Our strategy then is to solve for one of the barred variables via
\propref{prop:complexificationofrasurface}, and hope
\propref{prop:complexificationofrasurface} and hope
the CR conditions take care of the rest of the barred variables
in more than one dimension.

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