Proposition 16.6.2.8769D844C274 Given $\varphi ,\psi \in \mathrm{QPSH}(X)$, the following are equivalent:
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For any $k\in \mathbb {Z}_{>0}$, we have
\[ \mathcal{I}(k\varphi )=\mathcal{I}(k\psi ); \]
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for any $\lambda \in \mathbb {R}_{>0}$, we have
\[ \mathcal{I}(\lambda \varphi )=\mathcal{I}(\lambda \psi ); \]
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for any modification $\pi \colon Y\rightarrow X$ and any $y\in Y$, we have
\[ \nu (\pi ^*\varphi ,y)=\nu (\pi ^*\psi ,y); \]
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for any proper bimeromorphic morphism $\pi \colon Y\rightarrow X$ from a Kähler manifold and any $y\in Y$, we have
\[ \nu (\pi ^*\varphi ,y)=\nu (\pi ^*\psi ,y); \]
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for any prime divisor $E$ over $X$, we have
\[ \nu (\varphi ,E)=\nu (\psi ,E). \]
See definition 9.10.2 for the definition of prime divisors over $X$. Recall that here, a modification of a compact complex space means a finite composition of blow-ups with smooth centers. This terminology is highly non-standard.
For simplicity, we usually write
\[ \nu (\varphi ,y)=\nu (\pi ^*\varphi ,y) \]
when $\pi $ is obvious from the context.
Proof.
4 $\iff $ 5. This follows from 5.16.17.
3 $\iff $ 5. This follows from corollary 9.10.10.
1 $\implies $ 5. This follows from proposition 5.16.15.
5 $\implies $ 2. This follows from theorem 5.16.18.
2 $\implies $ 1. This is the restriction to positive integral values of $\lambda $.
$\square$
Definition 16.6.3.3B668848773E Given $\varphi ,\psi \in \mathrm{QPSH}(X)$, we say they are $\mathcal{I}$-equivalent and write $\varphi \sim _{\mathcal{I}}\psi $ if the equivalent conditions in proposition 16.6.2 are satisfied.
Clearly, $\sim _{\mathcal{I}}$ is an equivalence relation on $\mathrm{QPSH}(X)$.
Proposition 16.6.4.BBF7977859F1 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$ to $X$. For $\varphi ,\psi \in \mathrm{QPSH}(X)$, the following are equivalent:
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$\varphi \sim _{\mathcal{I}}\psi $;
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$\pi ^*\varphi \sim _{\mathcal{I}}\pi ^*\psi $.
Proof.
1 $\implies $ 2. This follows from proposition 16.6.2 4.
2 $\implies $ 1. This follows from the simple fact that
\[ \mathcal{I}(k\varphi )=\pi _* \left( \omega _{Y/X}\otimes \mathcal{I}(k\pi ^*\varphi ) \right),\quad \mathcal{I}(k\psi )=\pi _* \left( \omega _{Y/X}\otimes \mathcal{I}(k\pi ^*\psi ) \right) \]
for any $k\in \mathbb {Z}_{>0}$, where $\omega _{Y/X}$ is the relative dualizing sheaf. See [Proposition 5.8, Dem12].
$\square$
Proposition 16.6.5.26D170DCA6A1 Let $\varphi ,\varphi ',\psi ,\psi '\in \mathrm{QPSH}(X)$ and $\lambda >0$. Assume that $\varphi \sim _{\mathcal{I}}\psi $ and $\varphi '\sim _{\mathcal{I}}\psi '$. Then
\[ \varphi \lor \varphi '\sim _{\mathcal{I}} \psi \lor \psi ',\quad \varphi +\varphi '\sim _{\mathcal{I}} \psi +\psi ',\quad \lambda \varphi \sim _{\mathcal{I}}\lambda \psi . \]
Similarly, if $(\varphi _ i)_{i\in I}$, $(\psi _ i)_{i\in I}$ are two non-empty uniformly bounded from above families in $\mathrm{PSH}(X,\theta )$ for some closed smooth real $(1,1)$-form $\theta $ on $X$ such that $\varphi _ i\sim _{\mathcal{I}}\psi _ i$ for all $i\in I$, then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i\sim _{\mathcal{I}}\operatorname *{sup}_{i\in I}{}\! {}^{*} \psi _ i. \]
Proof.
This follows from proposition 5.16.8 and corollary 5.16.9.
$\square$
Pome
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