Proposition 16.8.7

Proposition 16.8.7.C7C64ABB4647 Let $\theta $ be a closed smooth real $(1,1)$-form on $X$ representing a big cohomology class. Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$ and $\varphi \preceq _ P \psi $. Then the following are equivalent:

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$\varphi \sim _ P \psi $;
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for each $j=0,\ldots ,n$, we have

DF3EF73A9B89 16.8.8
\begin{equation} \int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\psi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{equation}

Assume furthermore that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$, then these conditions are equivalent to the following:

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We have

\[ \int _ X \theta _{\varphi }^ n=\int _ X \theta _{\psi }^ n. \]

Proof. We first prove the equivalence between 1 and 1 when $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$.

1 $\implies $ 1. Assume that $\varphi \sim _ P \psi $. By 16.8.4, we have

\[ P_{\theta }[\varphi ]=P_{\theta }[\psi ]. \]

So 1 follows from proposition 16.3.10.

1 $\implies $ 1. Since $\varphi \preceq _ P\psi $, we have $P_{\theta }[\varphi ]\leq P_{\theta }[\psi ]$. Moreover, proposition 16.3.10 shows that $P_{\theta }[\varphi ]$ and $P_{\theta }[\psi ]$ have the same positive mass. By theorem 16.3.15, $P_{\theta }[\psi ]$ is a candidate in the characterization of $P_{\theta }\bigl [P_{\theta }[\varphi ]\bigr ]=P_{\theta }[\varphi ]$. Hence $P_{\theta }[\psi ]\leq P_{\theta }[\varphi ]$, and therefore $P_{\theta }[\varphi ]=P_{\theta }[\psi ]$. So 1 follows.

Let us come back to the general case with $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$.

1 $\implies $ 2. Fix $j\in \{ 0,\ldots ,n\} $. We argue (16.8.8).

Take a Kähler form $\omega $ on $X$. By 16.8.4, for each $\epsilon >0$, we have

\[ P_{\theta +\epsilon \omega }[\varphi ]= P_{\theta +\epsilon \omega }[\psi ]. \]

It follows from proposition 16.3.10 that

\[ \begin{aligned} \int _ X \left(\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\psi \right)^ j\wedge \theta _{V_{\theta }}^{n-j}=& \int _ X \Bigl (\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta +\epsilon \omega }[\psi ] \Bigr )^ j\wedge \theta _{V_{\theta }}^{n-j}\\ =& \int _ X \Bigl (\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta +\epsilon \omega }[\varphi ] \Bigr )^ j\wedge \theta _{V_{\theta }}^{n-j} \\ =& \int _ X \left(\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi \right)^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{aligned} \]

Since the two extremes are both polynomials in $\epsilon $ by the multi-linearity of the non-pluripolar products proposition 13.11.4 6, we conclude that the same holds when $\epsilon =0$, that is, (16.8.8) holds.

2 $\implies $ 1. Assume (16.8.8) holds for all $j=0,\ldots ,n$. Then the same holds with $P_{\theta }[\psi ]$ in place of $\psi $, thanks to proposition 16.3.10. Therefore, for each $t\in (0,1)$, we have

\[ \int _ X \theta _{t\varphi +(1-t)V_{\theta }}^ n= \int _ X \theta _{tP_{\theta }[\psi ]+(1-t)V_{\theta }}^ n \]

by the binomial expansion. Furthermore, $t\varphi +(1-t)V_{\theta }\preceq tP_{\theta }[\psi ]+(1-t)V_{\theta }$. By the implication 1 $\implies $ 1, we have

\[ t\varphi +(1-t)V_{\theta }\sim _ P t P_{\theta }[\psi ]+(1-t)V_{\theta } \]

for each $t\in (0,1)$.

Fix a Kähler form $\omega $ on $X$. From the implication 1 $\implies $ 1, we have

\[ \int _ X (\theta +\omega )_{t\varphi +(1-t)V_{\theta }}^ n= \int _ X (\theta +\omega )_{tP_{\theta }[\psi ]+(1-t)V_{\theta }}^ n. \]

Since both sides are polynomials in $t$ by the multi-linearity of the non-pluripolar products proposition 13.11.4 6, the same holds when $t=1$. From the implication 1 $\implies $ 1 again, we have $\varphi \sim _ P P_{\theta }[\psi ]\sim _ P \psi $. $\square$

Proposition 16.8.7`C7C64ABB4647`

Comments

Proposition 16.8.7C7C64ABB4647

Comments

Proposition 16.8.7`C7C64ABB4647`