PomeExample 20.2.23.531FFD6D4B64 The trace operator is not continuous along increasing sequences. Let us consider the case $X=\mathbb {P}^2$ with coordinates $(z_1,z_2)$ on $\mathbb {C}^2\subseteq X$. Let $\omega _{\mathrm{{FS}}}$ denote the Fubini–Study metric. The subvariety $Y\cong \mathbb {P}^1$ is defined by $z_2=0$. Consider an increasing sequence $(\varphi _ j)_ j$ in $\mathrm{PSH}(X,\omega _{\mathrm{FS}})$, whose potentials near $(0,0)$ are given by
\[ \log |z_1|^2\lor \left( j^{-1}\log |z_2|^2\right)+\mathcal{O}(1). \]
The pointwise restriction of these potentials to $Y$ are given locally by
\[ \log |z_1|^2+\mathcal{O}(1). \]
On the other hand, locally
\[ \log |z_1|^2\lor \left( j^{-1}\log |z_2|^2\right)\to 0 \]
almost everywhere as $j\to \infty $. So the trace operator is not continuous along the sequence $(\varphi _ j)_ j$.
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