Lemma 5.27.29.E2307F7792A2 Let $n\geq 2$, let $U\subset \mathbb C^{n-1}$ be a domain, and let $v$ be a smooth plurisubharmonic function on $U$. Fix $\varepsilon ,R,\eta >0$ and $A\in \mathbb R$. Let $\chi _\eta $ be the convolution of $t\mapsto \max (t,0)$ with a smooth even probability density supported in $[-\eta ,\eta ]$, and put
\[ \begin{split} q(\zeta )& =\varepsilon \log (|\zeta |^2/R^2),\qquad T(\zeta ,z')=v(z')-A-q(\zeta ),\\ h(\zeta ,z')& =q(\zeta )+\chi _\eta (T(\zeta ,z')) \end{split} \]
for $\zeta \ne 0$. Then $h$ extends to a smooth plurisubharmonic function on $\mathbb C\times U$. Set
\[ w_\eta (t)=n\chi _\eta '(t)^{n-1}\chi _\eta ''(t),\qquad r_ t(z')=R\exp \bigl ((v(z')-A-t)/(2\varepsilon )\bigr ). \]
The function $w_\eta $ is nonnegative, supported in $[-\eta ,\eta ]$, and has integral one. For every $f\in C_ c(\mathbb C\times U)$,
97724F2A7D19
5.27.30
\begin{equation} \begin{split} \int f(\mathrm{dd}^{\mathrm{c}}h)^ n =\varepsilon \int _ U\biggl [\int _{-\eta }^{\eta }w_\eta (t) \int _0^{2\pi }f(r_ t(z')\mathrm e^{\mathrm i\vartheta },z') \frac{\mathrm d\vartheta }{2\pi }\, \mathrm dt\biggr ] (\mathrm{dd}^{\mathrm{c}}v)^{n-1}. \end{split} \end{equation}
Proof.
By convexity, $0\leq \chi _\eta '\leq 1$ and $\chi _\eta ''\geq 0$. The function $\chi _\eta $ vanishes on $(-\infty ,-\eta ]$ and equals $t$ on $[\eta ,\infty )$. Since $T\to +\infty $ as $\zeta \to 0$, we have $h=v-A$ near each point with $\zeta =0$, which gives the smooth extension. Off the hyperplane $\{ \zeta =0\} $, the function $q$ is pluriharmonic and
\[ \mathrm{dd}^{\mathrm{c}}h=\chi _\eta '(T)\mathrm{dd}^{\mathrm{c}}v +\chi _\eta ''(T)\, \mathrm dT\wedge \mathrm d^{\mathrm c}T\geq 0. \]
The second summand has rank at most one, and the first has rank at most $n-1$. Hence
\[ (\mathrm{dd}^{\mathrm{c}}h)^ n=n\chi _\eta '(T)^{n-1}\chi _\eta ''(T) \, \mathrm dT\wedge \mathrm d^{\mathrm c}T\wedge (\mathrm{dd}^{\mathrm{c}}v)^{n-1}. \]
The last factor has top degree in the variables $z'$, so the derivatives of $v$ in $\mathrm dT\wedge \mathrm d^{\mathrm c}T$ contribute nothing. Write $s=\log (|\zeta |^2/R^2)$. In polar coordinates, $\mathrm dq=\varepsilon \, \mathrm ds$ and $\mathrm d^{\mathrm c}q=\varepsilon \, \mathrm d\vartheta /(2\pi )$. At fixed $z'$, the change of variables from $s$ to $t=v-A-\varepsilon s$ gives (5.27.30). There is no contribution near $\zeta =0$, where $h=v-A$ and the form $\mathrm{dd}^{\mathrm{c}}h$ has rank at most $n-1$. Finally, $w_\eta $ is the derivative of $(\chi _\eta ')^ n$, whose values at $-\eta $ and $\eta $ are zero and one.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.