Proposition 20.2.2258627BFD2457

Proof. It is enough to prove the sequential assertion. Indeed, suppose that the net statement fails. Then there is $\delta >0$ such that for every $i_0\in I$ one can find $i\geq i_0$ with

Using also $\varphi _ i\xrightarrow {d_ S}\varphi $, we may choose an increasing sequence $i_ j\in I$ such that

The sequence $(\varphi _{i_ j})_ j$ is decreasing and converges to $\varphi $ in $d_ S$, contradicting the sequential assertion.

Thus we assume that $(\varphi _ j)_{j\geq 1}$ is a decreasing sequence and that $\varphi _ j\xrightarrow {d_ S}\varphi $.

Take a Kähler form $\omega $ on $X$ so that $\varphi _ j,\varphi \in \mathrm{PSH}(X,\omega )_{>0}$. Replacing $\varphi _ j$ by $P_{\omega }[\varphi _ j]$, we may then assume that $(\varphi _ j)_ j$ is a decreasing sequence of model potentials. By proposition 16.9.47, we have

By proposition 16.9.6, we may assume from now on that

After adding a Kähler form to $\omega $, we may assume that $\omega _{\varphi }$ is a Kähler current.

For each $j\geq 1$, choose a quasi-equisingular approximation $(\varphi _ j^ k)_{k\geq 1}$ of $\varphi _ j$ in $\mathrm{PSH}(X,\omega )$. Using the standard construction in theorem 5.18.11, we may choose these approximations compatibly in $j$, so that for each fixed $k$,

We shall choose an increasing sequence of integers $k_ j$ and put

The sequence $(k_ j)_ j$ is chosen inductively: Choose $k_1>1$, for $j>1$, we choose $k_ j$ large enough so that

1. 1BA5628DE09C

   $k_{j}\geq k_{j-1}$;

2. C590440C99EE

   $d_{S,\omega }(\eta _ j,P_{\omega }[\varphi _ j]_{\mathcal{I}})\leq 1/(j-1)$;

3. 5061729EE0BF

   $d_{S,\omega |_{\widetilde{Y}}}(\eta _ j|_{\widetilde{Y}},\operatorname{Tr}_ Y^{\omega }(\varphi _ j))<1/(j-1)$.

It follows from the compatibility in $j$ and from $k_ j\geq k_{j-1}$ that $(\eta _ j)_ j$ is decreasing in singularity types. Indeed,

By theorem 16.9.49, we have

By 2 and corollary 18.1.14, we have $\eta _ j\xrightarrow {d_ S} P_{\omega }[\varphi ]_{\mathcal{I}}$. Observe that by proposition 16.9.47,

By proposition 16.6.22, for each $j>0$, $P_{\omega }[\eta _ j]$ has analytic singularities. By theorem 16.9.59, $(P_{\omega }[\eta _ j])_ j$ is a quasi-equisingular approximation of $P_{\omega }[\varphi ]_{\mathcal{I}}$. Therefore, by definition of the trace operator and proposition 20.2.21 1,

It follows from 3 that

We conclude the proof. $\square$