The $(1,1)$-form defined on $(\mathbb{C}\backslash\{0\})^{d}$ by

$$\eta:=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\Phi^{\mathrm{sm}}\left(\log|z|\right)$$

extends to a smooth (in the orbifold sense) non-negative definite $(1,1)$-form on $X_{T}$ in the cohomology class $[\eta]=[\nu]$.