First, we rewrite the form $\eta$ on $(\mathbb{C}\backslash\{0\})^{d}$ as

where ”$\langle\ \wedge\ \rangle$” means also the $\langle\ ,\ \rangle$-pairing between the $(\mathbb{R}^{d})^{*}$-valued gradient $\nabla\Phi^{\mathrm{sm}}$ and the $\mathbb{R}^{d}$-valued 1-form $d\operatorname{Arg}(z)$.

For a simplex $\tau\in T$ we want to show that $\eta$ extends to the toric subvariety $Z_{\tau}$ associated to $\tau$. If $X_{T}$ is smooth, then in a neighborhood of $Z_{\tau}$ we can choose the coordinates similar to those from [HZ02, Lemma 3.9]. That is, we choose a basis $\{e_{i}\}$ such that

Then, in the coordinates $y_{i}=z^{e_{i}}$ the equations for the subvariety $Z_{\tau}\subset X_{T}$ are $y_{i}=0,i=1,\dots,\dim\tau+1$.

According to the theory of toric varieties (cf., e.g. [Ful93]) a neighborhood of the toric subvariety $Z_{\tau}$ lies in the closure of $\log^{-1}(R_{\geq\tau})$, where $R_{\geq\tau}$ is any translation of the cone $\cup_{\tau^{\prime}\geq\tau}\operatorname{cone}(\tau^{\prime})$. But by the Lemma 3.3 the directional derivatives $\langle\nabla\Phi^{\mathrm{sm}},w_{i}\rangle$, $w_{i}\in\tau$, are constant in some translation $R_{\geq\tau}$ of $\cup_{\tau^{\prime}\geq\tau}\operatorname{cone}(\tau^{\prime})$. Hence, in a neighborhood of $Z_{\tau}$ the form $\eta_{a}$ written in the above coordinates is independent of $y_{i},i=1,\dots,\dim\tau+1$, and, thus, can be extended to $Z_{\tau}$.

In case when $X_{T}$ is an orbifold we may not be able to choose an integral basis $\{e_{i}\}$ with the above conditions. This corresponds to the fact that we may need to go to a finite cover to get a smooth form by weakening the first set of conditions to be $\langle e_{i},\tau\rangle\in\mathbb{Z}$. But the rest of the argument goes through.

Finally, the cohomology class of a $\mathbb{T}$-invariant $(1,1)$-form on a complete toric variety is determined by the image of its moment map. But the moment map for $\eta$ is given on $(\mathbb{C}\backslash\{0\})^{d}$ by

whose extension to the whole toric variety $X_{T}$ has the image ${\Delta_{\nu}}$. Hence the class of $\eta$ is $[\nu]$. ∎