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Corollary 20.1.36BF517753D33B

Corollary 20.1.36.BF517753D33B Let $T$ and $T'$ be non-divisorial closed positive $(1,1)$-currents on $X$. Suppose that $\{ T\} =\{ T'\} $. Then the following are equivalent:

  1. 5014A568828A

    $\mathbb {D}(T)\leq \mathbb {D}(T')$;

  2. 16D6081BA03D

    $T\preceq _{\mathcal{I}}T'$.

Proof. Let $E$ be a prime divisor over $X$. Choose a sequence of blow-ups with smooth connected centers as in the proof of theorem 20.1.28, so that $E$ appears as one of the exceptional prime divisors on the resulting model $\pi \colon Y\rightarrow X$. By (20.1.30), the difference

\[ \mathbb {D}(T')_ Y-\mathbb {D}(T)_ Y \]

is represented in the relative cohomology by

\[ \sum _ i\left(\nu (T,E_ i)-\nu (T',E_ i)\right)\{ E_ i\} . \]

Using proposition 20.1.2 inductively, or equivalently the negativity lemma for exceptional classes, pseudo-effectivity of this exceptional class is equivalent to the non-negativity of all coefficients. Hence $\mathbb {D}(T)\leq \mathbb {D}(T')$ is equivalent to

\[ \nu (T,E)\geq \nu (T',E) \]

for every prime divisor $E$ over $X$, which is exactly $T\preceq _{\mathcal{I}}T'$ by proposition 16.8.9. $\square$

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