Proposition 20.2.32AFC4F39F65CF

Proof. We start with the first equality of (20.2.33). Since $Y\not\subseteq \operatorname{nK}(\alpha )$, $V_{\theta }|_ Y \not\equiv -\infty $ as a consequence of 13.13.8, hence also $\nu (V_{\theta },Y)=0$.

Take a quasi-equisingular approximation $(\varphi _ j)_{j>0}$ of $V_{\theta }$ with $\varphi _ j\in \mathrm{PSH}(X,\theta +\epsilon _ j\omega )$ for a decreasing sequence $(\epsilon _ j)_{j>0}$ converging to $0$. By the monotonicity theorem theorem 13.13.11, we have for each $j>0$,

for every $\chi \in \mathrm{PSH}(X,\theta )$ with $\chi |_ Y\not\equiv -\infty $. Since $\chi $ is arbitrary, we conclude that

For any fixed $\epsilon >0$, we have

Letting $\epsilon \to 0+$ and applying example 20.2.17, we conclude that the $\leq $ direction in the first equality of (20.2.33) holds. Moreover, $Y\not\subseteq \operatorname{nK}(\alpha )$ implies $\operatorname{vol}_{X|Y}(\alpha )>0$ by (20.2.30). Hence the limiting mass in example 20.2.17 is positive, and $\operatorname{Tr}_ Y^\theta (V_\theta )$ is indeed defined in the present situation.

For the reverse direction, by definition, for any fixed $\epsilon >0$, we have

for all large enough $j$. Therefore, for large $j$, by the monotonicity theorem theorem 13.13.11, we have

Letting $j\to \infty $ and $\epsilon \searrow 0$, using the continuity of $\operatorname{vol}_{X|Y}$ ([Corollary 4.11, Mat13]) together with example 20.2.17, we conclude the first equality of (20.2.33).

Now we address the second equality. Due to theorem 13.13.11, the defining formula (20.2.30), and the definition of $V_\theta $, we obtain that

The reverse inequality now follows from the first equality of (20.2.33), theorem 13.13.11 and the fact that $V_\theta |_ Y \preceq _ P \operatorname{Tr}_ Y^\theta (V_\theta )$ as proved in proposition 20.2.12. $\square$