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22.1.78 The transcendental setting1630658F75F9

Definition 22.1.79.322D4668B967 For $\varphi \in \mathrm{PSH}(X,\theta )$, we define the analytification $\varphi ^{\mathrm{An}}\colon X^{\mathrm{An}}\rightarrow [-\infty ,0]$ as follows:

2CFF9C7C56DB 22.1.80
\begin{equation} \varphi ^{\mathrm{An}}(v)\coloneqq -v(\varphi )=-\lim _{k\to \infty }\frac{1}{k}v\left(\mathcal{I}(k\varphi ) \right). \end{equation}
By theorem 5.16.14 and Fekete's lemma, the limit in (22.1.80) exists.

Note that we can also write

DFFDD4E75B4D 22.1.81
\begin{equation} \varphi ^{\mathrm{An}}(v)=\inf _{k\in \mathbb {Z}_{>0}} -2^{-k}v\left( \mathcal{I}(2^ k\varphi )\right). \end{equation}

When $v=t\operatorname{ord}_ E$ for some prime divisor $E$ over $X$, $\varphi ^{\mathrm{An}}(v)=-t\nu (\varphi ,E)$ by proposition 5.16.15.

Definition 22.1.82.866728488224 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. We define the analytification $\Gamma ^{\mathrm{An}}\colon X^{\operatorname{div}}\rightarrow [-\infty ,\infty )$ of $\Gamma $ as follows: For any $\omega \in \textup{Käh}(X)$, we define

45957E422015 22.1.83
\begin{equation} \Gamma ^{\mathrm{An}}(v)\coloneqq \sup _{\tau \leq \Gamma _{\max }}\left(\Gamma _{\tau }^{\theta +\omega ,\mathrm{An}}(v)+\tau \right). \end{equation}
Clearly, (22.1.83) is independent of the choice of $\omega $.

Note that (22.1.83) can be equivalently written as

\[ \Gamma ^{\mathrm{An}}(v)= \sup _{\tau \leq \Gamma _{\max }}\left(\Gamma _{\tau }^{\theta +\omega ,\mathrm{An}}(v)+\tau \right)=\sup _{\tau \in \mathbb {R}}\left(\Gamma _{\tau }^{\theta +\omega ,\mathrm{An}}(v)+\tau \right) \]

with $(-\infty )^{\mathrm{An}}(v) = -\infty $ understood.

Proposition 22.1.84.A87B7BF0C3D5 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ with $\Gamma _{\max }\leq 0$. Let $\Psi $ be the complexification of $\Gamma ^*$. Then

B5F5AD50A5AD 22.1.85
\begin{equation} \Gamma ^{\mathrm{An}}(v)=-\sigma (v)(\Psi )\quad \forall v\in X^{\operatorname{div}}. \end{equation}
See definition 17.10.17 for the definition of the complexification $\Psi \in \mathrm{QPSH}(X\times \Delta )$. Note that since $\Gamma _{\max }\leq 0$, by corollary 17.13.31 and theorem 5.14.10, $\Psi $ extends uniquely to a quasi-psh function on $X\times \Delta $.

Proof. Recall that

\[ \Psi (x,\delta )=\sup _{\tau \leq \Gamma _{\max }} (\Gamma _{\tau }(x)-\log |\delta |^2\tau )\quad \text{for }x\in X, \delta \in \Delta ^*. \]

By (22.1.69), we have $\sigma (v)(\log |\delta |^2)=1$ and $\sigma (v)(\Gamma _{\tau })=v(\Gamma _{\tau })$ for all $\tau \leq \Gamma _{\max }$. So we have

\[ \sigma (v)(\Gamma _\tau (x) - \log |\delta |^2 \tau ) = v(\Gamma _\tau ) - \tau . \]

Lastly, since $\sigma (v)$ is a divisorial valuation on $X \times \Delta $, by corollary 5.16.9, we conclude (22.1.85). $\square$

Definition 22.1.86.54C62141C0D9 Let $N\in \mathbb {N}$, and $A_{0},\ldots ,A_ N$ be a finite collection of elements in $\mathrm{PSH}(X,\theta )$, and $\tau _0>\tau _1>\cdots >\tau _ N$ be finitely many real numbers. Then the piecewise linear curve $A=(A_{\tau })_{\tau \in \mathbb {R}}$ in $\mathrm{PSH}(X,\theta )\cup \{ -\infty \} $ associated with these data is the affine interpolation of these data:

  1. FE3BE40BF529

    $A_{\tau _ i}=A_ i$ for $i=0,\ldots ,N$;

  2. 2D3D174A0348

    $A_{\tau }=A_{\tau _ N}$ for $\tau \leq \tau _ N$;

  3. B0B44E5310EA

    for $t\in (0,1)$ and $i=0,\ldots ,N-1$, we have

    \[ A_{(1-t)\tau _ i+t\tau _{i+1}}=(1-t)A_{\tau _ i}+tA_{\tau _{i+1}}; \]
  4. 8FF2573D4019

    $A_{\tau }\equiv -\infty $ for $\tau >\tau _0$.

The analytification of $A$ is the function $A^{\mathrm{An}}\colon X^{\mathrm{An}}\rightarrow [-\infty ,\infty )$ defined as follows:

2B233D0F0F88 22.1.87
\begin{equation} A^{\mathrm{An}}(v)\coloneqq \sup _{\tau \leq \tau _0} (A_\tau ^{\mathrm{An}}(v)+\tau )=\max _{i=0,\ldots ,N} \left(A_{\tau _ i}^{\mathrm{An}}(v)+\tau _ i\right)\quad \forall v\in X^{\mathrm{An}}. \end{equation}
We also say $A=(A_{\tau })_{\tau \leq \tau _0}$ is a piecewise linear curve in $\mathrm{PSH}(X,\theta )$.

Remark 22.1.88.9911A537C86C Note that $\tau \mapsto A_{\tau }$ is upper semicontinuous, but not necessarily concave. Let $(A'_{\tau })_{\tau \in \mathbb {R}}$ be the upper concave envelope of $\tau \mapsto A_{\tau }$. Then it can be inductively constructed as follows:

  1. D954EE804570

    For $\tau \in (\tau _0,\infty )$, we let $A'_{\tau }\equiv -\infty $;

  2. 34859AD723A6

    we set $A'_{\tau _0}=A_{\tau _0}$;

  3. FCE6B95F3C1F

    define inductively for $j=0,\ldots ,N-1$ the following: For $\tau \in [\tau _{j+1},\tau _ j)$, we set

    \[ A'_{\tau }=\max _{i=j+1,\ldots ,N}\left( \frac{\tau _ j-\tau }{\tau _ j-\tau _ i}A_{\tau _ i}+\frac{\tau -\tau _ i}{\tau _ j-\tau _ i}A'_{\tau _ j} \right)\lor A'_{\tau _ j}; \]
  4. E47ACD33228B

    for $\tau \in (-\infty ,\tau _ N)$, we set $A'_{\tau }=A_{\tau _ N}$.

This construction is just a reformulation of the general formula proposition 21.11.12.

In particular, $A'_{\tau }\in \mathrm{PSH}(X,\theta )$ for all $\tau \leq \tau _0$.

Note that $A'$ is not necessarily piecewise linear.

Lemma 22.1.89.6784060E951F Let $A$ be a piecewise linear curve in $\mathrm{PSH}(X,\theta )$. Let $(A'_{\tau })_{\tau \in \mathbb {R}}$ be the upper concave envelope of $\tau \mapsto A_{\tau }$. Define $\tilde A$ by

\[ \tilde{A}^{\theta +\omega }_{\tau }\coloneqq P_{\theta +\omega }[A'_{\tau }]_{\mathcal I}, \quad \omega \in \textup{Käh}(X),\ \tau <\tau _0. \]

Then $\tilde A\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, and

F46F1129051C 22.1.90
\begin{equation} A^{\mathrm{An}}=\tilde{A}^{\mathrm{An}}\quad \text{on }X^{\operatorname{div}}. \end{equation}
Here $\tau _0$ is as in definition 22.1.86.

Proof. We continue to use the notations in definition 22.1.86. The fact that $\tilde{A}\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ follows from 22.1.88 and the cocycle property of the $\mathcal{I}$-projection. In order to prove (22.1.90), we fix $v\in X^{\operatorname{div}}$ and $\omega \in \textup{Käh}(X)$. The inductive construction in 22.1.88 uses only finite maxima and positive affine combinations of the potentials $A_{\tau _ i}$. On divisorial valuations, analytification commutes with both operations. Moreover, $P_{\theta +\omega }[\bullet ]_{\mathcal I}$ does not change divisorial valuations. Hence

\[ \tau \mapsto \left(P_{\theta +\omega }[A'_\tau ]_{\mathcal I}\right)^{\mathrm{An}}(v)= (A'_\tau )^{\mathrm{An}}(v) \]

is the ordinary upper concave envelope of $\tau \mapsto A_\tau ^{\mathrm{An}}(v)$. The Legendre-type supremum over $\tau $ is therefore unchanged by passing from $A$ to $A'$, which proves (22.1.90). $\square$

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