Theorem 19.6.24.74133E3920A2 Let $X$ be a connected compact Kähler manifold and $\theta $ be a closed smooth real $(1,1)$-form on $X$ representing a big cohomology class $\alpha $. Let $Y_{\bullet }$ be a smooth flag on $X$ and $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$. Then the map
\[ (-\infty ,\Gamma _{\max })\ni \tau \mapsto \Delta _{Y_{\bullet }}(\theta ,\Gamma )_{\tau }\coloneqq \Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau }) \]
defines an Okounkov test curve relative to $\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty })$.
If furthermore $\Gamma \in \mathrm{TC}^1(X,\theta ;\Gamma _{-\infty })$ (resp. $\mathrm{TC}^{\infty }(X,\theta ;\Gamma _{-\infty })$), then we have $\Delta _{Y_{\bullet }}(\theta ,\Gamma )\in \mathrm{TC}^1(\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty }))$ (resp. $\mathrm{TC}^{\infty }(\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty }))$).
Proof.
Consider $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$. We need to verify that $\Delta _{Y_{\bullet }}(\theta ,\Gamma )$ is an Okounkov test curve relative to $\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty })$.
First observe that $\tau \mapsto \Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau })$ is concave and decreasing for $\tau <\Gamma _{\max }$. The decreasing property follows from theorem 19.5.26 2. For concavity, use the valuative characterization theorem 19.5.40 together with the additivity of valuations in proposition 19.3.9: if $x\in \Delta _{Y_\bullet }(\theta ,\Gamma _\tau )$ and $y\in \Delta _{Y_\bullet }(\theta ,\Gamma _{\tau '})$, then $tx+(1-t)y$ is obtained from the current $tS+(1-t)S'$, which is more singular than $\Gamma _{t\tau +(1-t)\tau '}$. Hence
\[ t\Delta _{Y_\bullet }\left(\theta ,\Gamma _\tau \right)+(1-t)\Delta _{Y_\bullet }\left(\theta ,\Gamma _{\tau '}\right)\subseteq \Delta _{Y_\bullet }\left(\theta ,\Gamma _{t\tau +(1-t)\tau '}\right). \]
Next we show that as $\tau \to -\infty $, we have
\[ \Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau })\xrightarrow {d_{\mathrm{Haus}}}\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty }). \]
By theorem 19.5.26 2, the bodies $\Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau })$ are contained in $\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty })$ and increase as $\tau \to -\infty $. Thus it suffices to compute their volumes:
\[ \begin{split} \lim _{\tau \to -\infty }\operatorname{vol}\Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau })=\frac{1}{n!}\lim _{\tau \to -\infty }\operatorname{vol}(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{\tau })=\frac{1}{n!}\operatorname{vol}(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{-\infty }) \\ =\operatorname{vol}\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty }), \end{split} \]
where we applied theorem 19.5.26 and theorem 16.9.57.
When $\Gamma \in \mathrm{TC}^{\infty }(X,\theta ;\Gamma _{-\infty })$, it is clear that $\Delta _{Y_{\bullet }}(\theta ,\Gamma )\in \mathrm{TC}^{\infty }(\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty }))$.
When $\Gamma \in \mathrm{TC}^{1}(X,\theta ;\Gamma _{-\infty })$, by theorem 19.5.26 1, (17.12.12) and (19.6.2), we have
\[ \mathbf{E}^{\Gamma _{-\infty }}(\Gamma )=\mathbf{E}(\Delta _{Y_{\bullet }}(\theta ,\Gamma )). \]
Indeed, by theorem 19.5.26 1,
\[ n!\operatorname{vol}\Delta _{Y_{\bullet }}(\theta ,\Gamma _\tau )=\int _ X\theta _{\Gamma _\tau }^ n \]
for every $\tau <\Gamma _{\max }$, and the same identity holds at $\tau =-\infty $. Comparing (17.12.12) with (19.6.2) gives the asserted equality of energies.
So $\Delta _{Y_{\bullet }}(\theta ,\Gamma )\in \mathrm{TC}^1(\Delta _{Y_{\bullet }}(\theta ,\Gamma _{-\infty }))$.
$\square$
Pome
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