PomeRemark 19.5.43.A8051310CBAB We expect that the closure operation in (19.5.41) is not necessary. This problem is closely related to the Dirichlet problem of the trace operator, see Page for more details.
Remark 19.5.43A8051310CBAB
Proof. The $\supseteq $ direction in (19.5.41) follows from corollary 19.5.39 together with theorem 19.5.26 2.
Let us write
\[ D_{Y_{\bullet }}(T)=\left\{ \nu _{Y_{\bullet }}(S):S\in \mathcal{Z}_+(X,\alpha ),S\preceq _{\mathcal{I}}T\right\} \]
for the time being.
Step 1. Assume that $T$ has analytic singularities. We have
\[ \begin{split} \Delta _{Y_{\bullet }}(T)\supseteq & \overline{D_{Y_{\bullet }}(T)}\\ \supseteq & \overline{\left\{ \nu _{Y_{\bullet }}(S):\mathcal{Z}_+(X,\alpha )\ni S \textup{ has gentle analytic singularities}, S\preceq T\right\} }. \end{split} \]
It follows easily from theorem 19.5.4 that the volume of the right-hand side is equal to the volume of $\Delta _{Y_{\bullet }}(T)$, so (19.5.41) holds.
Step 2. Assume that $T$ is a Kähler current. Take a quasi-equisingular approximation $T_ j\in \mathcal{Z}_+(X,\alpha )$ of $T$. Next we use the language of psh functions. Let $\varphi _ j,\varphi \in \mathrm{PSH}(X,\theta )$ be the potentials corresponding to $T_ j,T$ for each $j\geq 1$.
Fix an integer $N>0$. For large enough $j\geq 1$, we can find $\psi _ j\in \mathrm{PSH}(X,\theta )_{>0}$ such that
\[ P_{\theta }[\varphi ]_{\mathcal{I}}\geq (1-N^{-1})\varphi _ j+N^{-1} \psi _ j. \]
The existence of $\psi _ j$ follows from 13.13.30. It follows that
\[ \begin{aligned} D_{Y_{\bullet }}(T)\supseteq & D_{Y_{\bullet }}\left(\theta +\mathrm{dd}^{\mathrm{c}}\left((1-N^{-1})\varphi _ j+N^{-1} \psi _ j\right)\right)\\ \supseteq & (1-N^{-1})D_{Y_{\bullet }}(T_ j)+N^{-1} D_{Y_{\bullet }}(\theta +\mathrm{dd}^{\mathrm{c}}\psi _ j). \end{aligned} \]
We claim that
\[ \overline{D_{Y_{\bullet }}(T)}\supseteq \bigcap _{j=1}^{\infty }\overline{D_{Y_{\bullet }}(T_ j)}. \]
Let $x$ be a point in the right-hand side. Fix $N>0$. For all large $j$, choose
\[ y_ j\in D_{Y_{\bullet }}(T_ j),\quad z_ j\in D_{Y_{\bullet }}(\theta +\mathrm{dd}^{\mathrm{c}}\psi _ j) \]
with $y_ j\to x$. By the preceding inclusion,
\[ (1-N^{-1})y_ j+N^{-1}z_ j\in D_{Y_{\bullet }}(T). \]
The points $x,y_ j,z_ j$ all lie in the compact set $\Delta _{Y_{\bullet }}(\alpha )$ after taking closures, by corollary 19.5.39. Hence, letting $j\to \infty $ and then $N\to \infty $, we get $x\in \overline{D_{Y_{\bullet }}(T)}$, proving the claim. Therefore, by Step 1, we conclude that
\[ \Delta _{Y_{\bullet }}(T)=\bigcap _{j=1}^{\infty }\overline{\Delta _{Y_{\bullet }}(T_ j)}=\bigcap _{j=1}^{\infty }\overline{D_{Y_{\bullet }}(T_ j)}\subseteq \overline{D_{Y_{\bullet }}(T)}. \]
The reverse direction is already known.
Step 3. Finally, consider the general case. Take a Kähler current $T'\in \mathcal{Z}_+(X,\alpha )$ more singular than $T$ (the existence of which is given by 13.13.47). For each $\epsilon \in (0,1)$, we know that
\[ \Delta _{Y_{\bullet }}((1-\epsilon )T+\epsilon T')=\overline{D_{Y_{\bullet }}((1-\epsilon )T+\epsilon T')} \subseteq \overline{D_{Y_{\bullet }}(T)}. \]
Letting $\epsilon \to 0+$ and using the $d_ S$-continuity in theorem 19.5.26 4, we find that
\[ \Delta _{Y_{\bullet }}(T)\subseteq \overline{D_{Y_{\bullet }}(T)}. \]
As the other inclusion is already known, we conclude. $\square$
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