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Example 17.13.32E2332A16AF00

Example 17.13.32.E2332A16AF00 Let us see what the test curve in example 17.12.13 corresponds to under the Ross–Witt Nyström correspondence. Fix $\varphi \in \mathrm{PSH}(X,\theta )$. We claim that

D50F33814868 17.13.33
\begin{equation} \ell ^{\varphi }=\Gamma ^{\varphi *}, \end{equation}

where $\ell ^{\varphi }$ is as in example 17.10.92. We may assume that $\varphi \leq 0$ since both sides are invariant after adding a constant to $\varphi $.

We first prove the easy direction $\ell ^{\varphi }\geq \Gamma ^{\varphi *}$, which is equivalent to $\ell ^{\varphi *}\geq \Gamma ^{\varphi }$. Since $\ell ^{\varphi *}$ is a test curve, the latter is equivalent to

\[ \ell ^{\varphi *}_{\tau }\geq (1+\tau )\varphi -\tau V_{\theta } \]

for all $\tau \in (-1,0)$. By Legendre duality, this is equivalent to

F5A9BEF8F7BA 17.13.34
\begin{equation} \ell ^{\varphi }_ t\geq \sup _{\tau \in (-1,0)} \Bigl ((1+\tau )\varphi -\tau V_{\theta }+t\tau \Bigr )=\varphi \lor (V_{\theta }-t) \end{equation}

for all $t>0$.

Using the notations of example 17.10.92, we find easily that

\[ \ell ^{\varphi ,C}_ t\geq \varphi \lor (V_{\theta }-t) \]

for any $C>0$ and $t\in [0,C]$, since it holds at $t=0$ and $t=C$. Letting $C\to \infty $, we find (17.13.34). Therefore, $\ell ^{\varphi }\geq \Gamma ^{\varphi *}$ follows.

In order to prove the equality in (17.13.33), it suffices to show that the two sides have the same energy, as a consequence of (17.10.84). So we compute

\[ \begin{aligned} \mathbf{E}\left(\Gamma ^{\varphi *}\right)=& \mathbf{E}\left(\Gamma ^{\varphi }\right)\\ =& \int _{-1}^0 \left(\int _ X \theta _{(1+\tau )\varphi -\tau V_{\theta }}^ n-\int _ X \theta _{V_{\theta }}^ n\right)\, \mathrm{d}\tau \\ =& \sum _{j=0}^{n} \binom {n}{j} \int _ X \theta _{\varphi }^{j}\wedge \theta _{V_{\theta }}^{n-j} \int _0^1 \tau ^ j (1-\tau )^{n-j}\, \mathrm{d}\tau -\int _ X \theta _{V_{\theta }}^ n\\ =& \sum _{j=0}^{n} \binom {n}{j}\frac{j!(n-j)!}{(n+1)!} \int _ X \theta _{\varphi }^{j}\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{V_{\theta }}^ n\\ =& \mathbf{E}(\ell ^{\varphi }), \end{aligned} \]

where in the second line we used proposition 16.3.10, in the fourth line we used the value of the $\beta $-function2, and the last line is just (17.10.94).

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