The discriminant locus $D=\Pi(\tau)\times\{0\}$ is of codimension 3 in $R^{n}\times\mathbb{C}$ and has a nice simplicial stratification. Using this stratification the topological compactification from $M$ to $\bar{M}=X_{\mathcal{T}}$ follows by extending the argument of [Gro01, Prop. 2.9] to arbitrary dimensions and including the orbifold singularities. To prove the matching of complex structure we will follow closely [LeB91] where the argument is given for the $\mathbb{C}^{2}$ case.

Let $\frac{\partial}{\partial\theta_{j}}$ be the Hamiltonian vector fields generating the $T^{n}$-action on $M$. Consider the commuting vector fields

where

denote the horizontal lifts of the $\frac{\partial}{\partial u_{i}}$, and $J:TM\to TM$ is the complex structure. Since the $\frac{\partial}{\partial\theta_{j}}$ preserve both the metric and the complex structure, it follows that the $\xi_{j}$ are holomorphic vector fields. Moreover, the their flows are complete because

hence, the $\xi_{j}$ generate a holomorphic action of $(\mathbb{C}^{*})^{n}$ on $\bar{M}$.

The orbit structure of this action is easily seen to be identical with the toric $(\mathbb{C}^{*})^{n}$-action on the orbifold $X_{\mathcal{T}}$. Namely, for each subspace

the set $\pi^{-1}(L_{a})$ is a union of orbits. For $a\neq 0$ the $\pi^{-1}(L_{a})$ is a single orbit, where as the $\pi^{-1}(L_{0})$ decomposes into $(n+1)$ orbits isomorphic to $(\mathbb{C}^{*})^{n}$ and a bunch of smaller dimensional orbits according to the polyhedral decomposition of $N^{*}_{\mathbb{R}}/\rho$ induced by $\Pi(\tau)$.

Consider the following subsets of $N^{*}_{\mathbb{R}}/\rho\times\mathbb{C}$:

We can cover the space $M$ by $(n+1)$ open sets $\mathcal{U}_{i}:=\pi^{-1}(L_{i})$. Each map $\eta:\mathcal{U}_{i}\to\mathbb{C}$ defines a holomorphic principal $(\mathbb{C}^{*})^{n}$-bundle over $\mathbb{C}$. Since $\mathbb{C}$ is Stein and contractible we conclude that each $\mathcal{U}_{i}$ is biholomorphic to the product $(\mathbb{C}^{*})^{n}\times\mathbb{C}$.

Now notice that the bundle structures on $\mathcal{U}_{i}$ agree on their intersection $\pi:\bigcap_{i}\mathcal{U}_{i}\to\mathbb{C}^{*}$. Thus, $M$ is biholomorphic to the quotient space

where the equivalence relation is of the form

for some cocycle $f_{ij}$ with values in the holomorphic maps $\mathbb{C}^{*}\to\operatorname{Aut}(\mathbb{C}^{*})^{n}$. But any automorphism of the principal homogeneous space $(\mathbb{C}^{*})^{n}$ is given by an $n$-tuple of non-zero complex numbers. In other words, each $f_{ij}$ is an $n$-tuple of holomorphic functions $\mathbb{C}^{*}\to\mathbb{C}^{*}$. Another choice of trivializations $\mathcal{U}_{i}\cong(\mathbb{C}^{*})^{n}\times\mathbb{C}$ amounts to modifying the cocycle $f_{ij}$ by a coboundary. That is, the biholomorphism type of $M$ is determined by the singularities of $f_{ij}(\eta)$ at $\eta=0$.

Next we notice that if any of the $f_{ij}(\eta)$ had an essential singularity at 0, then it would be possible to find a sequence of points in $M$ converging to several distinct points. Hence $M$ would not even be Hausdorff. Hence, every singularity has to be removable. But the orders of vanishing of the $f_{ij}(\eta)$ can be read off from the residues of the 1-forms $d\log f_{ij}$, and this is a topological information given by the Chern class of the original $T^{n}$-bundle. THus, the claimed biholomorphism is established on $M$, and it can be extended to $\bar{M}$ by an orbifold version of the Hartog’s theorem.

In order to show that the holomorphic volume form is necessary the standard one we will prove the following (stronger) statement. Given two sets of commuting vector fields $\xi_{i}$ and $\zeta_{i}$ generating the $(\mathbb{C}^{*})$-actions on $\eta:M\to\mathbb{C}$ which agree topologically, there is a biholomorphism $\phi:M\to M$ such that $\phi_{*}\xi_{i}=\zeta_{i}$. Then, taking $X_{i}$ to be the standard action on $X_{\mathcal{T}}$ yields the claim about the volume form.

Let $\{e_{1},\dots,e_{n}\}\in N^{*}$ together with $\rho$ form a basis of $N^{*}$ topologically compatible with $\xi_{i}$ and $\zeta_{i}$. That is, if $\{\alpha_{i},\eta\}$ and $\{\beta_{i},\eta\}$ are the two sets of coordinates, such that in the respective coordinates $\xi_{j}=\alpha_{j}\frac{\partial}{\partial\alpha_{j}}$ and $\zeta_{j}=\beta_{j}\frac{\partial}{\partial\beta_{j}}$ on the big toric orbit $\{\eta\neq 0\}\cong(\mathbb{C}^{*})^{n+1}$, then the transition maps between these coordinates and the standard ones $\{z^{e_{i}},\eta\}$ extends from $(\mathbb{C}^{*})^{n+1}$ to the whole of $M$ without zeros/poles. As a consequence, if we write

then the matrix $B_{j}^{k}$ is invertible and extends to $M$.

The goal is to find a change of coordinates $\beta_{j}=\alpha_{j}e^{\psi_{j}}$ which would induce such a Jacobian matrix. Using the chain rule

this amounts to solving the differential system

which, in general, is overdetermined.

However, in our case there are several restrictions on $B_{j}^{k}$. Namely, note that the forms

are closed and have to satisfy the periodicity requirements of the action:

where $\gamma_{k}$ is the generating cycle in the $k$-th factor of $\mathbb{C}^{*}$ in $(\mathbb{C}^{*})^{n}$. Then, by writing $B^{k}_{j}$ in the power series form

we conclude that $(b^{k}_{j})_{m,r}=0$ for $m_{j}=0$, except for $(b^{k}_{j})_{0,0}=\delta_{j}^{k}$, and $(b^{k}_{j})_{m,r}=0$ if there exists an $i\neq j$ with $m_{i}\neq 0$. Hence, one can simultaneously solve the equations

for all $m,r,k,j$, and write down the power series solution to (14):

that converges on the same polydisk in $M$ as the power series (15) for $B^{k}_{j}$ does. These functions $\psi^{k}$ provide the desired coordinate change. ∎