Pome

Theorem 21.13.8438F3E4DFCD04

Theorem 21.13.84.38F3E4DFCD04 Take $T$-invariant big divisors $D_1,\ldots ,D_ n$ on $X$. Consider $T_ c$-invariant closed positive $(1,1)$-currents $T_ i$ in $c_1(\mathcal{O}_ X(D_ i))$ for $i=1,\ldots ,n$. Then

862A2BCBD614 21.13.85
\begin{equation} n! \operatorname{vol}\Bigl ( \Delta (T_1),\ldots ,\Delta (T_ n) \Bigr )=\operatorname{vol}\left(T_1,\ldots ,T_ n \right). \end{equation}

Proof. It suffices to prove the following assertion:

8E6671867292 21.13.86
\begin{equation} \Delta (T_1+T_2)=\Delta (T_1)+\Delta (T_2).{}^{\href{#sgpt-a0000106283}{7}}\end{equation}

In fact, assuming this, by induction, for each tuple of non-negative integers $C_1,\ldots ,C_ n$, we have

\[ \Delta \left( \sum _{i=1}^ n C_ i T_ i \right)=\sum _{i=1}^ n C_ i \Delta (T_ i). \]

Comparing the volumes, we have

\[ \operatorname{vol}\Delta \left( \sum _{i=1}^ n C_ i T_ i \right)=\sum _{i_1,\ldots ,i_ n=1}^ n C_{i_1}\cdots C_{i_ n} \operatorname{vol}\left(\Delta \left(T_{i_1}\right),\ldots ,\Delta \left(T_{i_ n}\right)\right). \]

On the other hand, by proposition 21.13.28, the $\mathcal{I}$-goodness of toric singularities example 18.1.56, and proposition 18.1.27, we have

\[ n! \operatorname{vol}\Delta \left( \sum _{i=1}^ n C_ i T_ i \right)=\operatorname{vol}\left( \sum _{i=1}^ n C_ i T_ i \right)=\sum _{i_1,\ldots ,i_ n=1}^ n C_{i_1}\cdots C_{i_ n} \operatorname{vol}\left(T_{i_1},\ldots ,T_{i_ n}\right). \]

Comparing the coefficients of the above two expressions, we conclude (21.13.85).

It only remains to establish (21.13.86). For this purpose, we fix smooth $T_ c$-invariant Hermitian metrics $h_1$ and $h_2$ on $\mathcal{O}_ X(D_1)$ and $\mathcal{O}_ X(D_2)$ respectively. Correspondingly, take $h_1\otimes h_2$ as the reference Hermitian metric on $\mathcal{O}_ X(D_1+D_2)$. Write

\[ \theta _1=\mathrm{dd}^{\mathrm{c}}h_1,\quad \theta _2=\mathrm{dd}^{\mathrm{c}}h_2. \]

We choose the rational section of $\mathcal O_ X(D_1+D_2)$ to be $s_{D_1}\otimes s_{D_2}$. With this choice and with the reference metric $h_1\otimes h_2$, the normalization (21.13.23) gives

\[ F_{\theta _1+\theta _2}=F_{\theta _1}+F_{\theta _2}. \]

Thus

\[ F_{\varphi _1+\varphi _2}=F_{\varphi _1}+F_{\varphi _2}. \]

Take $\varphi _ i\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta _ i)$ with $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ for $i=1,2$. Then $F_{\varphi _1}+F_{\varphi _2}$ is a candidate for $F_{\varphi _1+\varphi _2}$; see (21.13.26). Therefore, (21.13.86) follows from the equality of closed gradient images proved in 21.11.47. $\square$

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