Theorem 19.4.35.10DA5B8BC320 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Then for any $t\in (0,1)$,
E375090F5B43
19.4.36
\begin{equation} \Delta _{\nu }\left(\theta ,t\varphi +(1-t)\psi \right)\supseteq t\Delta _{\nu }(\theta ,\varphi )+(1-t)\Delta _{\nu }(\theta ,\psi ). \end{equation}
Proof.
We may assume that $t$ is rational as a consequence of theorem 19.4.30. Similarly, as in the proof of proposition 19.4.33, we may reduce to the case where both $\varphi $ and $\psi $ have analytic singularities. In this case, let $N>0$ be an integer such that $Nt$ is an integer. Then for any $s\in \mathrm{H}^0(X,L^ k\otimes \mathcal{I}_{\infty }(k\varphi ))$ and $r\in \mathrm{H}^0(X,L^ k\otimes \mathcal{I}_{\infty }(k\psi ))$, we have
\[ s^{tN}\otimes r^{N-tN}\in \mathrm{H}^0\left(X,L^{kN}\otimes \mathcal{I}_{\infty }\left(kN(t\varphi +(1-t)\psi )\right)\right). \]
By theorem 19.2.5 and 19.4.23, (19.4.36) follows.
$\square$
Pome
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