Proposition 20.2.4

Proposition 20.2.4.82453E358B30 Let $\varphi \in \mathrm{QPSH}(X)$ be a function such that $\nu (\varphi ,Y)=0$. Let $(\varphi _ i)_ i$, $(\psi _ i)_ i$ be quasi-equisingular approximations of $\varphi $. Then

2B91479A9BE3 20.2.5

\begin{equation} \lim _{i\to \infty } d_ S\left(\varphi _ i|_{\widetilde{Y}},\psi _ i|_{\widetilde{Y}}\right)=0. \end{equation}

Proof. After subtracting a common constant from $\varphi $, $\varphi _ i$ and $\psi _ i$, we may assume that $\varphi _ i\leq 0$ for all $i\geq 1$.

Take a Kähler form $\omega $ on $X$ such that $\varphi _ i,\psi _ i\in \mathrm{PSH}(X,\omega /2)$ for all $i\geq 1$. By corollary 16.9.36, we need to show that

\[ \lim _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}}\left(\varphi _ i|_{\widetilde{Y}},\psi _ i|_{\widetilde{Y}}\right)=0. \]

Assume that this fails. Then, up to replacing the sequences by subsequences, we may assume that the following limit exists and

\[ \lim _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}}\left(\varphi _ i|_{\widetilde{Y}},\psi _ i|_{\widetilde{Y}}\right)>0. \]

Take a Kähler form $\tilde{\omega }$ on $\widetilde{Y}$. Then

\[ \lim _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}+\tilde{\omega }}\left(\varphi _ i|_{\widetilde{Y}},\psi _ i|_{\widetilde{Y}}\right)>0 \]

by corollary 16.9.36.

By corollary 18.1.14, after changing the background class from $\omega /2$ to $\omega $ if necessary, we have

\[ \varphi _ i\xrightarrow {d_ S}P_{\omega }[\varphi ]_{\mathcal I},\quad \psi _ i\xrightarrow {d_ S}P_{\omega }[\varphi ]_{\mathcal I}. \]

Let

\[ \chi _ i\coloneqq \varphi _ i\lor \psi _ i. \]

By proposition 16.9.42, we have $\chi _ i\xrightarrow {d_ S} P_{\omega }[\varphi ]_{\mathcal{I}}$. Moreover, $(\chi _ i)_ i$ is again a quasi-equisingular approximation of the original potential $\varphi $ as follows from corollary 18.1.14.

The triangle inequality gives

\[ d_ S(\varphi _ i|_{\widetilde{Y}},\psi _ i|_{\widetilde{Y}}) \leq d_ S(\varphi _ i|_{\widetilde{Y}},\chi _ i|_{\widetilde{Y}})+ d_ S(\psi _ i|_{\widetilde{Y}},\chi _ i|_{\widetilde{Y}}). \]

Thus, after passing to a subsequence and interchanging the two quasi-equisingular approximations if necessary, we may assume that

\[ \varliminf _{i\to \infty }d_ S(\varphi _ i|_{\widetilde{Y}},\chi _ i|_{\widetilde{Y}})>0. \]

Replacing $(\psi _ i)_ i$ by $(\chi _ i)_ i$, we may therefore assume that

\[ \varphi _ i\leq \psi _ i \]

for all $i$, while preserving the contradiction assumption.

Take a decreasing sequence $(\epsilon _ j)_ j$ in $\mathbb {R}_{>0}$ with limit $0$ such that $(1-\epsilon _ j)\varphi _ j\in \mathrm{PSH}(X,\omega )$. We first observe that

\[ \lim _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}}\left(\varphi _ i|_{\widetilde{Y}},(1-\epsilon _ i)\varphi _ i|_{\widetilde{Y}}\right)=0. \]

This is a consequence of 16.9.25. Hence, by corollary 16.9.36, we find

\[ \lim _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}+\tilde{\omega }}\left(\varphi _ i|_{\widetilde{Y}},(1-\epsilon _ i)\varphi _ i|_{\widetilde{Y}}\right)=0. \]

Since $(\varphi _ i)_ i$ and $(\epsilon _ i)_ i$ are decreasing and $\varphi _ i\leq 0$, the sequence $\bigl ((1-\epsilon _ i)\varphi _ i|_{\widetilde{Y}}\bigr )_ i$ is decreasing. It has uniformly positive mass in the class $\omega |_{\widetilde{Y}}+\tilde{\omega }$. Hence corollary 16.9.24, together with the preceding estimate, gives a potential $\psi \in \mathrm{PSH}(\widetilde{Y},\omega |_{\widetilde{Y}}+\tilde{\omega })$ such that

\[ \varphi _ i|_{\widetilde{Y}}\xrightarrow {d_ S}\psi . \]

Hence,

\[ \lim _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}+\tilde{\omega }}\left(\psi ,(1-\epsilon _ i)\varphi _ i|_{\widetilde{Y}}\right)=0. \]

By proposition 5.18.12, after choosing $j_ i$ inductively with $j_ i\geq i$ and $j_ i>j_{i-1}$, we may assume that

\[ \psi _{j_ i}\preceq (1-\epsilon _ i)\varphi _ i. \]

Since we have arranged $\varphi _ j\leq \psi _ j$ for all $j$, it follows that

\[ \varphi _{j_ i}\leq \psi _{j_ i}\preceq (1-\epsilon _ i)\varphi _ i. \]

Hence,

\[ \varphi _{j_ i}|_{\widetilde{Y}}\leq \psi _{j_ i}|_{\widetilde{Y}}\preceq (1-\epsilon _ i)\varphi _ i|_{\widetilde{Y}}. \]

Therefore, by corollary 16.9.8,

\[ \begin{aligned} \varlimsup _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}+\tilde{\omega }}\left(\varphi _{j_ i}|_{\widetilde{Y}},\psi _{j_ i}|_{\widetilde{Y}}\right) \leq & \varlimsup _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}+\tilde{\omega }}\left(\varphi _{j_ i}|_{\widetilde{Y}},(1-\epsilon _ i)\varphi _ i|_{\widetilde{Y}}\right)\\ =& \varlimsup _{i\to \infty } d_{S,\omega |_{\widetilde{Y}}+\tilde{\omega }}\left(\psi ,(1-\epsilon _ i)\varphi _ i|_{\widetilde{Y}}\right)\\ = & 0, \end{aligned} \]

which is a contradiction. $\square$

Proposition 20.2.4`82453E358B30`

Comments

Proposition 20.2.482453E358B30

Comments

Proposition 20.2.4`82453E358B30`