PomeExample 20.2.2.482E170F0825 Consider the case where $\varphi $ has log-log singularities as in example 16.3.22. Locally we can take $\varphi (z)=-\log (-\log |z|^2)$. We have $\nu (\varphi ,0)=0$ but $\varphi |_0=-\infty $. So the naive restriction is not defined in this case.
20.2.1 The definition of the trace operatorE3F1F75C84E9
Let $X$ be a connected compact Kähler manifold and $Y\subseteq X$ be an irreducible analytic subset.
The trace operator gives a way to restrict a quasi-plurisubharmonic function on $X$ to $\widetilde{Y}$, the normalization of $Y$. It follows from [Proposition 3.6, GK20] that $\widetilde{Y}$ is a normal Kähler space. We refer to section 9.10 for the pluripotential theory on unibranch Kähler spaces.
For later applications, we need this generality even if initially we are only interested in the smooth case.
Before diving into the trace operator, let us try to understand what goes wrong with the naive restriction procedure: simply take $\varphi |_{\widetilde{Y}}$. Both examples below are local, but can easily be globalized since the singularities are isolated.Example 20.2.3.DC6BE2600DA0 Consider a psh function $\varphi $ in two variables, say
\[ \varphi (z,w)=\left( -\log (-\log |z|^2)\right) \lor \left(\log |w|^2\right). \]
Take an arbitrary quasi-equisingular approximation $(\varphi _ j)_ j$. Then each $\varphi _ j$ is locally bounded since $\nu (\varphi ,(0,0))=0$. Let us consider the restrictions to $H=\{ z=0\} $. Then $\varphi _ j|_ H$ is still bounded, while
\[ \varphi |_ H(w)=\log |w|^2. \]
In other words, the restrictions $\varphi _ j|_ H$ fail to be a quasi-equisingular approximation of $\varphi |_ H$.
We first observe that if $\varphi \in \mathrm{QPSH}(X)$ has analytic singularities and $\nu (\varphi ,Y)=0$, then $\varphi |_ Y\not\equiv -\infty $. Indeed, if $\varphi |_ Y\equiv -\infty $, then in a local presentation
\[ \varphi =c\log (|f_1|^2+\cdots +|f_ N|^2)+\mathcal{O}(1) \]
all generators $f_1,\ldots ,f_ N$ vanish along $Y$, which gives $\nu (\varphi ,Y)>0$, a contradiction.
This observation will be crucial in the sequel.
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. $\square$
Definition 20.2.6.ADF6E265888F Let $\varphi \in \mathrm{QPSH}(X)$ be a function such that $\nu (\varphi ,Y)=0$. We say a potential $\psi \in \mathrm{QPSH}(\widetilde{Y})$ is a1 trace operator of $\varphi $ along $Y$ if there is a quasi-equisingular approximation $(\varphi _ j)_{j>0}$ of $\varphi $ such that
7AF21EBA8F4E
20.2.7
\begin{equation} \varphi _ j|_{\widetilde{Y}}\xrightarrow {d_ S}\psi {}^{\href{#sgpt-a0000065657}{2}}. \end{equation}
Later on in theorem 21.13.75, we shall prove that the trace operator corresponds to the natural way of restricting convex bodies in the toric setting.
Proposition 20.2.8.45EA06425921 Let $\varphi \in \mathrm{QPSH}(X)$ such that $\nu (\varphi ,Y)=0$. Suppose that $\psi $ and $\psi '$ are trace operators of $\varphi $ along $Y$. Then $\psi $ and $\psi '$ are $\mathcal{I}$-good and $\psi \sim _ P\psi '$.
Proof. The restrictions of the analytic approximants have analytic singularities on $\widetilde{Y}$, so the claim follows from theorem 18.1.2. The fact that $\psi \sim _ P\psi '$ follows from proposition 20.2.4 and proposition 16.9.6. $\square$
Example 20.2.9.75F5F5ACD2FAAs a trivial example, when $Y$ is just a single point, then $\mathrm{QPSH}(Y)$ is canonically identified with $\mathbb {R}$. Any constant $c\in \mathbb {R}$ is a trace operator of a function $\varphi \in \mathrm{QPSH}(X)$ satisfying $\nu (\varphi ,Y)=0$.
Definition 20.2.10.5239E9ADD0FF Let $\varphi \in \mathrm{QPSH}(X)$ such that $\nu (\varphi ,Y)=0$. We write $\operatorname{Tr}_{Y}(\varphi )$ for any trace operator of $\varphi $ along $Y$.
Given a closed smooth real $(1,1)$-form $\theta $ on $X$. When $\operatorname{Tr}_ Y(\varphi )$ can be chosen to lie in $\mathrm{PSH}(\widetilde{Y},\theta |_{\widetilde{Y}})_{>0}$, we write
\[ \operatorname{Tr}_ Y^{\theta }(\varphi )\coloneqq P_{\theta |_{\widetilde{Y}}}\bigl [ \operatorname{Tr}_ Y(\varphi ) \bigr ]=P_{\theta |_{\widetilde{Y}}}\bigl [ \operatorname{Tr}_ Y(\varphi ) \bigr ]_{\mathcal{I}}. \]
Remark 20.2.11.F5CDFD8547BC As in 5.19.5, the trace operator could also be applied to closed positive $(1,1)$-currents on $X$. If $T\in \mathcal{Z}_+(X,\alpha )$ for some pseudo-effective class $\alpha $ on $X$ (see definition 5.19.3) and $\beta \in \mathrm{H}^{1,1}(\widetilde{Y},\mathbb {R})$, then we write
\[ \operatorname{Tr}^{\beta }_ Y(T) \]
for any (if exists) closed positive $(1,1)$-current in $\beta $ representing $\operatorname{Tr}_ Y(T)$ when $\nu (T,Y)=0$.
Proposition 20.2.12.B293F74D2022 Let $\varphi \in \mathrm{QPSH}(X)$ such that $\nu (\varphi ,Y)=0$. Assume that $\varphi |_ Y\not\equiv -\infty $. Then
\[ \varphi |_{\widetilde{Y}}\preceq _ P \operatorname{Tr}_ Y(\varphi ). \]
Proof. Take a Kähler form $\omega $ such that $\omega _{\varphi }$ is a Kähler current. Let $(\varphi _ j)_{j>0}$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\omega )_{>0}$. We may assume that $\varphi _ j\leq 0$ for all $j\geq 1$.
Then
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20.2.13
\begin{equation} \varphi _ j|_{\widetilde{Y}}\leq P_{\omega |_{\widetilde{Y}}}\left[\varphi _ j|_{\widetilde{Y}}\right] \end{equation}
for all $j\geq 1$. In particular,
\[ \varphi |_{\widetilde{Y}}\leq \inf _{j\geq 1}P_{\omega |_{\widetilde{Y}}}\left[\varphi _ j|_{\widetilde{Y}}\right]. \]
Thanks to corollary 16.9.14,
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20.2.14
\begin{equation} \operatorname{Tr}_ Y(\varphi )\sim _ P \inf _{j\geq 1} P_{\omega |_{\widetilde{Y}}}[\varphi _ j|_{\widetilde{Y}}]. \end{equation}
We conclude our assertion. $\square$
Example 20.2.15.917FA516701F Let $\varphi \in \mathrm{QPSH}(X)$ such that $\nu (\varphi ,Y)=0$. Assume that $\varphi $ has analytic singularities. Then
\[ \operatorname{Tr}_ Y(\varphi )\sim _ P \varphi |_{\widetilde{Y}}. \]
Example 20.2.16.50B7DFD48C9D Let $\varphi \in \mathrm{QPSH}(X)$. Take a closed real smooth $(1,1)$-form $\theta $ on $X$ such that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, then
\[ \operatorname{Tr}_ X(\varphi )\sim _ P P_{\theta }[\varphi ]_{\mathcal{I}},\quad \operatorname{Tr}^{\theta }_ X(\varphi ) = P_{\theta }[\varphi ]_{\mathcal{I}}. \]
In particular, the trace operator can be regarded as a generalization of the $\mathcal{I}$-envelope.
Example 20.2.17.2272C87A04CE Assume that $\varphi \in \mathrm{PSH}(X,\theta )$ for some closed smooth real $(1,1)$-form $\theta $ on $X$, $\nu (\varphi ,Y)=0$ and
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20.2.18
\begin{equation} \lim _{\epsilon \searrow 0} \int _{\widetilde{Y}} \left( \theta |_{\widetilde{Y}} +\epsilon \omega |_{\widetilde{Y}}+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta +\epsilon \omega }(\varphi )\right)^{\dim Y} >0 \end{equation}
for any arbitrary choice of a Kähler form $\omega $ on $X$. Then it follows from proposition 16.4.4 that $\operatorname{Tr}_ Y^{\theta }(\varphi )$ is defined, and its mass is exactly the above limit.
As a consequence, we have the following formula:
8.9
\begin{equation} \int _{\widetilde{Y}} \left(\theta |_{\widetilde{Y}}+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta }(\varphi )\right)^{\dim Y}=\lim _{\epsilon \searrow 0} \int _{\widetilde{Y}} \left( \theta |_{\widetilde{Y}} +\epsilon \omega |_{\widetilde{Y}}+\mathrm{dd}^{\mathrm{c}}\operatorname{Tr}_ Y^{\theta +\epsilon \omega }(\varphi )\right)^{\dim Y}, \end{equation}
where the left-hand side is understood as $0$ if $\operatorname{Tr}_ Y^{\theta }(\varphi )$ is not defined.
Remark 20.2.19.ECCF7648C028 The trace operator allows us to introduce the following extension of the moving Seshadri constant. For $T\in \mathcal{Z}_+(X,\alpha )$ and $x\in X$, define
\[ \epsilon (T,x)\coloneqq \inf _{V\ni x}\left(\frac{\operatorname{vol}\operatorname{Tr}_ V^{\alpha |_{\widetilde{V}}} T}{\operatorname{mult}_ x V} \right)^{\frac{1}{\dim V}}, \]
where $\operatorname{vol}\operatorname{Tr}_ V^{\alpha |_{\widetilde{V}}} T=0$ if $\operatorname{Tr}_ V^{\alpha |_{\widetilde{V}}} T$ is not defined. Here $V$ runs over all positive-dimensional closed irreducible analytic subsets of $X$ containing $x$.
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