Pome

Proposition 14.12.3029A1D30A55E

Proposition 14.12.3.029A1D30A55E The function $\rho _ u$ of proposition 14.12.1 is characterized by

BD3712C889A6 14.12.4
\begin{equation} \mathcal A_\gamma (\rho _ u\eta _ u)=0, \qquad \int _ X\rho _ u\eta _ u^ n=1. \end{equation}

When $n=2$, its first equation is $\mathrm{dd}^{\mathrm{c}}(\rho _ u\eta _ u)=0$. When $\eta _ u=\gamma $, the function $\rho _ u$ is constant and $\mathrm{d}\mu _ u^\gamma =\gamma ^ n/\int _ X\gamma ^ n$.

Proof. We first prove that, for every smooth real $(1,1)$-form $A$ and every smooth real function $v$ on $X$,

195CC5F57ABE 14.12.5
\begin{equation} \int _ X A\wedge \mathcal D_\gamma v =\int _ X v\, \mathcal A_\gamma (A). \end{equation}

Write $c=\mathrm i/(2\pi )$. Since $A,K,v$ are real, expanding the real part gives

\[ \mathcal D_\gamma v =c\, \partial \bar\partial v\wedge K +\frac c2\bigl (\partial v\wedge \bar\partial K -\bar\partial v\wedge \partial K\bigr ). \]

Integrating the first term by parts twice gives $\int _ Xv\, c\partial \bar\partial (A\wedge K)$, and integrating each first-order term by parts once gives the correction

\[ \int _ Xv\, \frac c2 \bigl (-\partial A\wedge \bar\partial K +\bar\partial A\wedge \partial K -2A\wedge \partial \bar\partial K\bigr ). \]

Finally,

\[ \partial \bar\partial (A\wedge K) =\partial \bar\partial A\wedge K -\bar\partial A\wedge \partial K +\partial A\wedge \bar\partial K +A\wedge \partial \bar\partial K. \]

Adding these contributions cancels the terms containing $A\wedge \partial \bar\partial K$ and leaves $\mathcal A_\gamma (A)$. This proves (14.12.5). Applied to $A=\rho _ u\eta _ u$, it shows by (14.12.2) that $\int _ X\mathcal L_ uv\, \mathrm{d}\mu _ u^\gamma $ is a fixed positive multiple of $\int _ Xv\, \mathcal A_\gamma (\rho _ u\eta _ u)$, so the adjoint condition is equivalent to (14.12.4).

For $n=2$, we have $K=1$, which gives the first special case. For the second, the product rule gives

\[ \mathcal A_\gamma (\gamma ) =\frac1{n-1}\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0. \]

By uniqueness, $\rho _ u$ is consequently the constant $(\int _ X\gamma ^ n)^{-1}$. $\square$

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