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Proposition 11.5.4BFC1330527CA

Proposition 11.5.4.BFC1330527CA For any two Hermitian forms $\eta ,\gamma $,

1F99A7154650 11.5.5
\begin{equation} \eta \wedge \gamma ^{n-1} \geq \left(\frac{\eta ^ n}{\gamma ^ n}\right)^{1/n}\gamma ^ n \end{equation}

pointwise. Consequently, let $\theta $ be a smooth real $(1,1)$-form and $f$ a smooth strictly positive function. If $u\in C^\infty (X,\mathbb R)$ and $c>0$ satisfy $(\theta +\mathrm{dd}^{\mathrm{c}}u)^ n=c f\, \mathrm{d}V$ with $\theta +\mathrm{dd}^{\mathrm{c}}u>0$, then

B78F87EF7411 11.5.6
\begin{equation} c^{1/n} \int _ X\left(\frac{f\, \mathrm{d}V}{\gamma ^ n}\right)^{1/n}\gamma ^ n \leq \int _ X\theta \wedge \gamma ^{n-1} \end{equation}

for every Gauduchon metric $\gamma $.

Proof. In a frame where $\gamma $ is the identity, let $\lambda _1,\ldots ,\lambda _ n>0$ be the eigenvalues of $\eta $. The ratios of the two sides of (11.5.5) to $\gamma ^ n$ are $n^{-1}\sum _ j\lambda _ j$ and $(\prod _ j\lambda _ j)^{1/n}$, respectively, and their comparison is the arithmetic–geometric mean inequality.

For the second assertion, apply (11.5.5) with $\eta =\theta +\mathrm{dd}^{\mathrm{c}}u$, integrate, and use (11.5.2). $\square$

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