Corollary 21.13.73.AFC1919DC91A Let $\ell \in \mathcal{R}_{\mathrm{tor}}(X,\theta ;\phi )$ and $g$ be the function as in corollary 21.13.71. Then
FE80C413846E
21.13.74
\begin{equation} \ell ^*_{\max }=-\inf _{M_{\mathbb {R}}} g, \end{equation}
and for each $\tau <-\inf _{M_{\mathbb {R}}} g$, the function $\ell ^*_{\tau }$ is toric-invariant, and
\[ G_{\ell ^*_{\tau }}(m)=\begin{cases} G_{\phi }(m),& \textup{if }g(m)\leq -\tau ;\\ \infty ,& \textup{otherwise}. \end{cases} \]
In particular, for such $\tau $,
\[ \Delta \left(\theta , \ell ^*_{\tau }\right)=\left\{ g\leq -\tau \right\} . \]
Proof.
Fix $\tau \in \mathbb {R}$. Then
\[ \ell ^*_{\tau }=\inf _{t>0} \left(\ell _ t-t\tau \right) \]
is clearly toric-invariant. Therefore,
\[ F_{\ell ^*_{\tau }}=\inf _{t>0} \left(F_{\ell _ t}-t\tau \right). \]
Fix $m\in M_{\mathbb {R}}$. We compute
\[ \begin{aligned} G_{\ell ^*_{\tau }}(m)=& \sup _{n\in N_{\mathbb {R}}}\left( \langle m, n\rangle -F_{\ell ^*_{\tau }}(n) \right)\\ =& \sup _{t>0}\sup _{n\in N_{\mathbb {R}}}\left( \langle m, n\rangle - F_{\ell _ t}(n) +t\tau \right)\\ =& \sup _{t>0} \left(G_{\ell _ t}(m)+t\tau \right)\\ =& G_{\phi }(m)+\sup _{t>0} t\left(g(m)+\tau \right)\\ =& \begin{cases} G_{\phi }(m),& \textup{if }g(m)+\tau \leq 0;\\ \infty ,& \textup{otherwise}. \end{cases}\end{aligned} \]
Our assertions follow.
$\square$
Pome
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