Corollary 20.1.37926F05B89741

Proof. Take a Kähler form $\omega $ on $X$. By theorem 20.1.24, for each $i>0$, we can find a non-divisorial Kähler current $T_ i\in \mathbb {D}_ X+i^{-1}\{ \omega \} $ such that

We observe that $T_ i$ and $T_{i+1}$ have the same $\mathcal{I}$-singularity type. Indeed, applying corollary 20.1.36 to $T_ i$ and $T_{i+1}+(i^{-1}-(i+1)^{-1})\omega $ shows that these two currents are $\mathcal{I}$-equivalent; adding the smooth form does not change the singularity type. Let $(T_ i^ j)_ j$ be quasi-equisingular approximations of $T_ i$ such that

1. 7B39195D3D69

   $T_ i^ j$ is a Kähler current in $\mathbb {D}_ X+i^{-1}\{ \omega \} $ for $j\geq j_0(i)$, and

2. CD8709C3E97A

   for each fixed $j$, the singularity type of $T_ i^ j$ is independent of $i$.

Note that 2 is possible by using the Bergman kernel construction of the quasi-equisingular approximations, since the $T_ i$'s have the same multiplier ideals.

For fixed $j$, the b-divisor

is independent of $i$. Since the quasi-equisingular approximations are decreasing in $j$, the sequence $(\mathbb {B}_ j)_ j$ is decreasing, and proposition 20.1.19 gives $\mathbb {B}_ j\to \mathbb {D}$. It suffices to take $\mathbb {D}_ i=\mathbb {B}_{j_ i}+i^{-1}\mathbb {D}(\omega )=\mathbb {D}(T_ i^{j_ i})$, where $j_ i$ is chosen to be a sufficiently fast increasing sequence of positive integers with $j_ i\geq j_0(i)$. $\square$