For a given $(\lambda,\nu)$ we define subsets $U_{v}^{\beta}\subset U_{v}$ and $V_{w}^{\beta^{\vee}}\subset V_{w}$ whose union will give the complement of a neighborhood of $D$ depending on two real parameters $\beta,{\beta^{\vee}}>0$. We assume $\beta,{\beta^{\vee}}$ to be small in the $\lambda,\nu$ scales, respectively. In what follows we identify $\Sigma$ with $\partial\Delta^{\vee}_{\lambda}$ and $\partial{\Delta_{\nu}}$ via the maps $\phi$ and $\hat{\phi}$ associated with the bi-PIKAS of type $(\lambda,\nu)$.

For $v\in\operatorname{vert}(S)$ we let $U_{v}^{\beta}$ be the set of points in the corresponding facet of $\Delta^{\vee}_{\lambda}$ which lie in the closed polyhedron $Q^{\lambda}_{(v\mid\{0\})}(\beta)$ (cf. [HZ02, Section 3.2]), and similar for $V_{w}^{\beta^{\vee}}$. Explicitly,

	$$\displaystyle U_{v}^{\beta}:=\{n\in U_{v}\ :\ \langle m,n\rangle+\lambda(m)\leq-\beta,\text{ all }m\in\Delta_{\mathbb{Z}}\backslash\{v,0\}\},$$
	$$\displaystyle V_{w}^{\beta^{\vee}}:=\{m\in V_{w}\ :\ \langle m,n\rangle+\nu(n)\leq-{\beta^{\vee}},\text{ all }n\in\Delta^{\vee}_{\mathbb{Z}}\backslash\{w,0\}\}.$$

Because $\beta,\beta^{\vee}$ are small the sets $U_{v}^{\beta}$ and $V_{w}^{\beta^{\vee}}$ are non-empty. We define the smooth part of $\Sigma$ as

$$\Sigma^{\mathrm{sm}}:=\bigcup_{v\in\operatorname{vert}(S)}U_{v}^{\beta}\cup\bigcup_{w\in\operatorname{vert}(T)}V_{w}^{\beta^{\vee}}.$$

Then the neighborhood of $D$ is defined as the complement to all these closed sets in $\Sigma$:

$$N_{\lambda,\nu}^{\beta,\beta^{\vee}}(D):=\Sigma\backslash\Sigma^{\mathrm{sm}}.$$

An important observation is that $N_{\lambda,\nu}^{\beta,\beta^{\vee}}(D)\to D$ as $\beta,\beta^{\vee}\to 0$ in the scales of $\lambda,\nu$, respectively.

We will be smoothing various functions later on in this section so let us recall the standard regularization techniques. Given a convex bounded domain $P\subset\mathbb{R}^{n}$ with piece-wise smooth boundary let $\rho$ be a mollifier whose support is $P$, i.e. a positive $\mathbb{C}^{\infty}(\mathbb{R}^{n})$ function vanishing exactly outside $P$ and such that $\int_{\mathbb{R}^{n}}\rho dx=1$. For instance, if $P$ is a polytope in $\mathbb{R}^{n}$ given by a collection of inequalities $\{\langle v_{i},x\rangle+\lambda_{i}\leq 0\}$ one can take the usual bell-shaped function

$$\rho=c\prod_{i}\rho_{i},\quad\text{ where }\rho_{i}(x)=\begin{cases}e^{\frac{1}{\langle v_{i},x\rangle+\lambda_{i}}},&\quad\langle v_{i},x\rangle+\lambda_{i}\leq 0\\ 0,&\quad\langle v_{i},x\rangle+\lambda_{i}\geq 0\end{cases},$$

and the constant $c$ is determined from the normalization. Set $\rho_{h}:=\frac{1}{h^{n}}\rho(\frac{x}{h})$. For any locally integrable function $u\in L^{1}_{loc}(\mathbb{R}^{n})$ we can apply the standard regularization procedure by taking the convolution with $\rho_{h}$:

$$u_{h}(x):=\rho_{h}\ast u=\int_{\mathbb{R}^{n}}\rho_{h}(x-y)u(y)\,dy.$$

Then, if $u\in C^{p}(\mathbb{R}^{n})$, the functions $u_{h}$ are $\mathbb{C}^{\infty}$ and approaching $u$ as $h\to 0$ in the $C^{p}$-norm uniformly on any compact in $\mathbb{R}^{n}$.

Below we list some elementary properties of $u_{h}$ which will be useful later.

One can also apply the regularization procedure by taking the convolution with the mollifier parameter depending (smoothly) on the point $x\in\mathbb{R}^{d}$. That is $u_{h}(x):=\int_{\mathbb{R}^{n}}\rho_{h(x)}(x-y)u(y)\,dy$. Or, even, more generally the entire shape of the support of the mollifier can smoothly depend on the center of convolution. We have used this technique of varying support to prove the existence of bi-PIKAS in Proposition 2.1.

We will construct a foliation of $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda}$ by straight lines/rays by specifying a “convex” vector field on $\partial\Delta^{\vee}_{\lambda}$, which is smooth in $\Sigma^{\mathrm{sm}}$.

Recall that the piece-wise linear functions $L_{\nu}:(\mathbb{R}^{d})^{*}\to\mathbb{R}$ and $L_{\lambda}:\mathbb{R}^{d}\to\mathbb{R}$, the Legendre transforms of $\nu$ and $\lambda$, were defined as

$$L_{\nu}(m):=\max_{n\in\Delta^{\vee}_{\mathbb{Z}}}\{\langle m,n\rangle+\nu(n)\},\quad L_{\lambda}(n):=\max_{m\in\Delta_{\mathbb{Z}}}\{\langle m,n\rangle+\lambda(m)\}.$$

We fix a mollifier $\rho^{\vee}$ on $(\mathbb{R}^{d})^{*}$ with support in $\Delta$ and consider smooth functions $L_{\nu,h^{\vee}}:=\rho^{\vee}_{h^{\vee}}\ast L_{\nu-h^{\vee}}$. From the properties of regularization (for ${h^{\vee}}$ small in the $\nu$-scale) the slopes $\nabla L_{\nu,{h^{\vee}}}(x)$ always lie in $\partial\Delta^{\vee}$, for any $x$ not in the interior of ${\Delta_{\nu}}$. In particular, the gradient of $L_{\nu,{h^{\vee}}}$ gives a map $\nabla L_{\nu,{h^{\vee}}}:\partial{\Delta_{\nu}}\to\partial\Delta^{\vee}$.

Now we can use the identification of $\partial{\Delta_{\nu}}$ with $\partial\Delta^{\vee}_{\lambda}$ via $\phi\hat{\phi}^{-1}$ given by the $(\lambda,\nu)$-bi-PIKAS to define a $\partial\Delta^{\vee}$-valued vector field $\mathfrak{X}_{h^{\vee}}$ on $\partial\Delta^{\vee}_{\lambda}$ by

$$\mathfrak{X}_{h^{\vee}}(q):=\nabla L_{\nu,h^{\vee}}(\phi\hat{\phi}^{-1}q).$$

We can extend this vector field $\mathfrak{X}_{h^{\vee}}$ to $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda}$ using the following identification of $\partial{\Delta_{\nu}}$ with $\partial\Delta^{\vee}_{\lambda+t}$, for any $t\geq 0$. If $\hat{\Phi}$ is the (dual) $(\lambda,\nu)$-bi-PIKAS potential, then $\hat{\Phi}+tL_{\nu,{h^{\vee}}}$ is a strictly convex function on ${\Delta_{\nu}}$. In particular, its gradient defines a bijection $\nabla\hat{\Phi}+t\nabla L_{\nu,{h^{\vee}}}:\partial{\Delta_{\nu}}\to\partial\Delta^{\vee}_{\lambda+t}$. We can write the vector field $\mathfrak{X}_{h^{\vee}}$ on $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda}$ explicitly, using the fact that

$$L_{\lambda}(x)=t\Leftrightarrow x\in\partial\Delta^{\vee}_{\lambda+t},\text{ for any }t\geq 0.$$

Namely,

$$\mathfrak{X}_{h^{\vee}}(x):=\nabla L_{\nu,h^{\vee}}([\nabla\hat{\Phi}+L_{\lambda}(x)\nabla L_{\nu,{h^{\vee}}}]^{-1}(x)).$$

We summarize properties of this vector field in the following lemma:

First, we would like to extend the bi-PIKAS potential to $\mathbb{R}^{d}$ by taking the Legendre transform of $\hat{\Phi}$. Namely,

$$\Phi(x)=\max_{y\in{\Delta_{\nu}}}\{\langle x,y\rangle-\hat{\Phi}(y)\}.$$

Similarly, we extend $\hat{\Phi}$ to a function on $(\mathbb{R}^{d})^{*}$. We will abuse the notation $\Phi,\hat{\Phi}$ for the extended potentials.

$\Phi$ is a $C^{1}$-function, smooth when restricted to any strata of $\Delta^{\vee}_{\lambda}$. Its Hessian $\operatorname{Hess}\Phi$ is continuous at $\partial\mathcal{U}$, but blows off at $\partial\mathcal{V}$. And something drastic happens at the discriminant $D=\partial\mathcal{U}\cap\partial\mathcal{V}$.

Next we regularize the $C^{1}$-potential $\Phi$ to get a smooth convex function on $\mathbb{R}^{d}$ which we will use later on to define a Kähler potential on the toric variety $X_{T}$. Let $\rho$ be a mollifier with support in $-\Delta^{\vee}$. We define $\Phi^{\mathrm{sm}}_{h}:=\rho_{h}\ast\Phi.$

Before constructing a Kähler potential on the toric variety $X_{T}$ we need another technical statement.

Now we can use $\Phi^{\mathrm{sm}}=\Phi^{\mathrm{sm}}_{h}$ with any $h>0$ to define a Kähler potential on $X_{T}$. For an element $z=\{z_{1},\dots,z_{d}\}\in\mathbb{C}\backslash\{0\}^{d}$ we will use the notations

$$\log|z|:=\{\log|z_{1}|,\dots,\log|z_{d}|\}\in\mathbb{R}^{d},\quad\operatorname{Arg}(z):=\frac{1}{2\pi}\{\arg(z_{1}),\dots,\arg(z_{d})\}\in\mathbb{T}.$$

Finally we can add to $\eta$ a (small) positive multiple of a Kähler (e.g., Fubini-Study) form $\omega_{0}$. Thus we get a true Kähler form $\omega=\eta+\epsilon\omega_{0}$ on $X_{T}$ in the class $[\nu]+\epsilon[\omega_{0}])$.