Lemma 20.1.44.D57BEED0545E Let $\omega $ be a Kähler form on $X$. Fix a compact set $K\subseteq \mathrm{H}^{1,1}(X,\mathbb {R})$. Let $\mathbb {D}_1,\ldots ,\mathbb {D}_ n$ be nef b-divisors over $X$ such that $\mathbb {D}_{i,X}\in K$ for each $i=1,\ldots ,n$. Then there is a constant $C$ depending only on $X,K,\{ \omega \} $ such that for any $\epsilon \in [0,1]$, we have
\[ 0\leq \Bigl ( \mathbb {D}_1+\epsilon \mathbb {D}(\omega ),\ldots ,\mathbb {D}_ n+\epsilon \mathbb {D}(\omega ) \Bigr )-\left( \mathbb {D}_1,\ldots ,\mathbb {D}_ n \right)\leq C\epsilon . \]
Proof.
By 20.1.40 4, the left-hand side is non-negative. For the upper bound, by the multi-linearity of the product proved in proposition 20.1.43, we can rewrite the difference as a sum of $2^ n-1$ terms like
\[ \epsilon ^{m} \Bigl ( \mathbb {D}(\omega )^{m}, \mathbb {D}_{i_1},\ldots , \mathbb {D}_{i_{n-m}} \Bigr ), \]
where $J\subseteq \{ 1,\ldots ,n\} $ is a non-empty subset of cardinality $m$, and $\{ i_1,\ldots ,i_{n-m}\} $ is the complement of $J$. We need to show that these coefficients are uniformly bounded by a constant depending only on the given data. For this purpose, we add $\mathbb {D}(\omega )$ to each $\mathbb {D}_ i$ and then replace $K$ by $K+\{ \omega \} $. Therefore, we reduce to the case where each $\mathbb {D}_ i$ is nef and big, and hence $\mathbb {D}_ i=\mathbb {D}(T_ i)$ for some $\mathcal{I}$-good closed positive $(1,1)$-currents on $X$ with positive volumes. Then we need to bound
\[ \int _ X \omega ^ m\wedge T_{i_1}\wedge \cdots \wedge T_{i_{n-m}}. \]
For this purpose, let $D\geq 0$ be the minimal non-negative number so that $D\{ \omega \} -\alpha $ is nef for any $\alpha \in K$. Then the integral is bounded by the cohomological intersection $\{ \omega \} ^ m\cap (D\{ \omega \} )^{n-m}$. Our assertion follows.
$\square$
Pome
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