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Proposition 12.24.17CE9EDC18591

Proposition 12.24.1.7CE9EDC18591 Let $\theta ,\beta $ be smooth nef real $(1,1)$-forms on a compact complex manifold $X$ of dimension $n$, and suppose that

\[ S=(\theta -\beta )+\mathrm{dd}^{\mathrm{c}}\varphi \geq 0 \]

is a positive $(1,1)$-current for some integrable function $\varphi $. For every Hermitian form $h$, every smooth strictly positive probability form $\mu $ and every $\epsilon >0$,

2818CCA10129 12.24.2
\begin{equation} \mathcal V_\mu (\beta +\epsilon h)\leq \mathcal V_\mu (\theta +\epsilon h). \end{equation}

Consequently $\underline a(\beta )\geq \underline a(\theta )$ and $\overline a(\beta )\geq \overline a(\theta )$.

Proof. For every Gauduchon metric $\gamma $, distributional integration by parts gives

\[ \int _ X(\theta -\beta )\wedge \gamma ^{n-1} =\langle S,\gamma ^{n-1}\rangle \geq 0. \]

Hence, for each test metric, the quotient for $\theta +\epsilon h$ is at least the quotient for $\beta +\epsilon h$, and taking infima gives $\lambda _\mu (\beta +\epsilon h)\leq \lambda _\mu (\theta +\epsilon h)$. Both sides are positive because $\beta $ and $\theta $ are nef, so raising to the $n$th power gives (12.24.2). Dividing the logarithms by the negative number $\log \epsilon $ reverses the inequality and proves both comparisons of orders. $\square$

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