Proposition 16.8.9EE590FE41862

Proposition 16.8.9.EE590FE41862 Given $\varphi ,\psi \in \mathrm{QPSH}(X)$, the following are equivalent:

1. ABA41C3D6B24

   For any $k\in \mathbb {Z}_{>0}$, we have

   \[ \mathcal{I}(k\varphi )\subseteq \mathcal{I}(k\psi ); \]
2. C970FD27DA7D

   for any $\lambda \in \mathbb {R}_{>0}$, we have

   \[ \mathcal{I}(\lambda \varphi )\subseteq \mathcal{I}(\lambda \psi ); \]
3. EE2809B429C8

   for any modification $\pi \colon Y\rightarrow X$ and any $y\in Y$, we have

   \[ \nu (\pi ^*\varphi ,y)\geq \nu (\pi ^*\psi ,y); \]
4. F03432967BAB

   for any proper bimeromorphic morphism $\pi \colon Y\rightarrow X$ from a Kähler manifold and any $y\in Y$, we have

   \[ \nu (\pi ^*\varphi ,y)\geq \nu (\pi ^*\psi ,y); \]
5. 7B8BA50C8017

   for any prime divisor $E$ over $X$, we have

   \[ \nu (\varphi ,E)\geq \nu (\psi ,E). \]