Since the measure $c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}$ is given by a smooth volume form and $X^{{\text{\rm an}}}_{\Sigma}\setminus X^{{\text{\rm an}}}_{0}$ is a set of Lebesgue measure zero, the measure $c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}$ is determined by its restriction to the dense open subset $X_{0}^{{\text{\rm an}}}$. Thus, to prove equation (5.34) it is enough to show that

(5.35)

$${\operatorname{val}}_{\ast}(c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}|_{X^{{\text{\rm an}}}_{0}})=n!\mathcal{M}_{M}(\psi).$$

We use the coordinate system of the proof of Proposition 5.29. We denote by ${\widetilde{{\operatorname{\mathbf{e}}}}}\colon N_{\mathbb{C}}\to X_{0}(\mathbb{C})$ the map induced by the morphism $\mathbb{C}\to\mathbb{C}^{\times}$ given by $z\mapsto\exp(-z)$. We write $u_{k}+iv_{k}$ for the complex coordinates of $N_{\mathbb{C}}$. Then

(5.36)

$${\widetilde{{\operatorname{\mathbf{e}}}}}^{\ast}\left(\frac{\,\text{\rm d}z_{k}\land\,\text{\rm d}\bar{z}_{k}}{z_{k}\bar{z}_{k}}\right)=(-2i)\,\text{\rm d}u_{k}\land\,\text{\rm d}v_{k}.$$

Using now equations (5.30), (5.31) and (5.36), we obtain that,

	$\displaystyle\frac{1}{(2\pi i)^{n}}{\widetilde{{\operatorname{\mathbf{e}}}}}^{\ast}c_{1}({\overline{L}})^{n}$	$\displaystyle={\widetilde{{\operatorname{\mathbf{e}}}}}^{\ast}\left(\frac{1}{(i\pi)^{n}}n!\det G\,\text{\rm d}z_{1}\land\,\text{\rm d}\bar{z}_{1}\land\dots\land\,\text{\rm d}z_{n}\land\,\text{\rm d}\bar{z}_{n}\right)$
		$\displaystyle=\frac{(-1)^{n}}{(2\pi)^{n}}n!\det\operatorname{Hess}(\psi)\,\text{\rm d}u_{1}\land\,\text{\rm d}v_{1}\land\dots\land\,\text{\rm d}u_{n}\land\,\text{\rm d}u_{n}.$

Since the map ${\operatorname{val}}$ is the composition of ${\widetilde{{\operatorname{\mathbf{e}}}}}^{-1}$ with the projection $N_{\mathbb{C}}\to N_{\mathbb{R}}$, integrating with respect to the variables $v_{1},\dots,v_{n}$ in the domain $[0,2\pi]^{n}$, taking into account the natural orientation of $\mathbb{C}^{n}$ and the orientation of $N_{\mathbb{R}}$ given by the coordinate system, and the fact that the normalization factor $1/(2\pi i)^{n}$ is implicit in the current $\delta_{X_{\Sigma}}$, we obtain

$${\operatorname{val}}_{\ast}(c_{1}({\overline{L}})^{n}\land\delta_{X_{\Sigma}}|_{X_{0}^{{\text{\rm an}}}})=(-1)^{n}n!\det\operatorname{Hess}(\psi)\,\text{\rm d}u_{1}\land\dots\land\,\text{\rm d}u_{n}.$$

Thus equation (5.35) follows from Proposition 3.94. Finally, the last statement follows from the fact that, in a compact Abelian group there is a unique Haar measure with fixed total volume. ∎