Proposition 22.1.1099F673331DE01

Proof. Take a Kähler form $\omega $ on $X$. We need to show that

Therefore, after replacing $\theta $ by $\theta +\omega $, we may assume that $\Gamma _ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ for all $i\in I$ and $\inf _{i\in I}\Gamma _ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$. Fix $v\in X^{\operatorname{div}}$. By theorem 22.1.72, it suffices to prove that

Write $\Gamma =\inf _{i\in I}\Gamma _ i$. We first compare the two sides inside the Legendre-type supremum. If $\tau <\Gamma _{\max }$, then by the definition of the infimum test curve,

The net $(\Gamma _{i,\tau })_ i$ decreases to $\Gamma _{\tau }$. By 17.15.41, $\Gamma _{\tau }$ is stable under $P_{\theta }[\bullet ]$. Hence proposition 16.4.4 gives convergence of the masses, and corollary 16.9.14 yields

Hence theorem 16.9.55 gives

If $\tau >\Gamma _{\max }$, then $\Gamma _{\tau }\equiv -\infty $, and also $\inf _ i\Gamma _{i,\tau }\equiv -\infty $, since $\Gamma _{\max }=\inf _ i(\Gamma _ i)_{\max }$. Thus the same equality holds for $\tau >\Gamma _{\max }$ as well, with value $-\infty $. The possible endpoint $\tau =\Gamma _{\max }$ does not affect the supremum: for any test curve $\Lambda $ and any $v\in X^{\operatorname{div}}$,

because $\Lambda _{\Lambda _{\max }}\leq \Lambda _{\tau }$ for $\tau <\Lambda _{\max }$, hence

and we let $\tau \nearrow \Lambda _{\max }$.

Consequently, in (22.1.110) we may replace $\left(\inf _ i\Gamma _{i,\tau }\right)^{\mathrm{An}}(v)$ by $\inf _ i\Gamma _{i,\tau }^{\mathrm{An}}(v)$ inside the supremum. The equality then follows from proposition 21.11.36 applied to the closed convex functions $-\Gamma _{i,\bullet }^{\mathrm{An}}(v)$. $\square$