ScalingStacks

Example 5.2. When $n=m$, namely in the maximal degeneration case, for simplicity we consider an $(n-1)$-dimensional face $\Delta_{J}$ of $Sk(X)$, such that there are only two $n$-dimensional faces of $\Delta_{\mathcal{X}}$ containing $\Delta_{J}$, and they both lie on $Sk(X)$. Then the degree condition $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})\cdot E_{J}=0$ imposes a matching condition on the gradient of $\phi_{0}$ across the $(n-1)$-dim face.

We consider a very concrete local example where $\mathbb{P}^{1}\simeq E_{J}=\cap_{1}^{n}E_{i}$,

$$\deg\mathcal{O}(E_{i})|_{E_{J}}=-d_{i}\leq 0,\quad i=1,2,\ldots n,\quad\sum_{1}^{n}d_{i}=2,$$

and the two divisors $E_{0},E_{\infty}$ intersect $E_{J}$ transversely at $0,\infty\in\mathbb{P}^{1}$, giving rise to the two $n$-dimensional faces. All divisors are reduced. The complex geometric local model for $\mathcal{X}$ comprises of two charts $\mathbb{C}^{n+1}_{z_{0},\ldots z_{n}}$ and $\mathbb{C}^{n+1}_{w_{0},\ldots,w_{n}}$. On the first chart $E_{i}=\{z_{i}=0\}$ for $i=0,1,\ldots n$, and $z_{0}$ is the affine coordinate on $\mathbb{P}^{1}\setminus\{\infty\}$. On the second chart $E_{i}=\{w_{i}=0\}$ for $i=1,\ldots n$, $E_{\infty}=\{w_{0}=0\}$, and $w_{0}$ is the affine coordinate on $\mathbb{P}^{1}\setminus\{0\}$. The transition on the overlap is

$$w_{0}=z_{0}^{-1},\quad w_{i}=z_{i}z_{0}^{d_{i}},\quad i=1,2,\ldots n.$$

The holomorphic volume form $\Omega\sim dz_{0}\wedge\ldots dz_{n}=-dw_{0}\wedge\ldots dw_{n}$, and the coordinate $t=z_{0}\ldots z_{n}=w_{0}\ldots w_{n}$. Locally on $X_{t}$

$$\Omega_{t}\sim d\log z_{1}\wedge\ldots d\log z_{n}=-d\log w_{1}\wedge\ldots d\log w_{n}.$$

On the two $n$-dimensional faces the coordinates are $x_{i}=\frac{\log|z_{i}|}{\log|t|}$ and $x_{i}^{\prime}=\frac{\log|w_{i}|}{\log|t|}$ respectively, satisfying the linear constraints $\sum_{0}^{n}x_{i}=0$ and $\sum_{0}^{n}x_{i}^{\prime}=0$, so we can eliminate $x_{n},x_{n}^{\prime}$. By adjusting $\mathcal{L}$ we can make it zero in the local model. The potential $\phi_{0}$ satisfies the real MA equation on the two $n$-dimensional faces:

$$\det\left(\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\right)_{0\leq i,j\leq n-1}=\text{const},\quad\det\left(\frac{\partial^{2}\phi_{0}}{\partial x_{i}^{\prime}\partial x_{j}^{\prime}}\right)_{0\leq i,j\leq n-1}=\text{const},$$

and the problem is to match them on $\Delta_{J}$. The complex geometry suggests that the domain of the coordinates $x_{i},x_{i}^{\prime}$ can be extended outside the original $n$-simplices, by the identificaton

$$x_{0}^{\prime}=-x_{0},\quad x_{i}^{\prime}=x_{i}+d_{i}x_{0},\quad i=1,2,\ldots n-1,$$

and the real MA equation is satisfied also on $\Delta_{J}$. In terms of gradients at $x\in\Delta_{J}$,

$$\frac{\partial\phi_{0}}{\partial x_{i}^{\prime}}=\frac{\partial\phi_{0}}{\partial x_{i}},\quad i=1,\ldots n-1,\quad\frac{\partial\phi_{0}}{\partial x_{0}^{\prime}}=-\frac{\partial\phi_{0}}{\partial x_{0}}+\sum_{1}^{n-1}d_{i}\frac{\partial\phi_{0}}{\partial x_{i}}.$$

The first $(n-1)$ conditions merely mean the tangent derivatives along $\Delta_{J}$ agree. The normal derivative matching condition precisely says

$$\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)\cdot E_{J}=\sum_{1}^{n-1}d_{i}\frac{\partial\phi_{0}}{\partial x_{i}}-\frac{\partial\phi_{0}}{\partial x_{0}}-\frac{\partial\phi_{0}}{\partial x_{0}^{\prime}}=0.$$

From a different perspective, the zero degree condition explains why $\mathbb{P}^{1}$ can be invisible to the metric, so $\phi_{0}$ is allowed to remain smooth across $\text{Int}(\Delta_{J})$.

Example 5.3. For $m=1$, consider a $0$-dimensional face $\Delta_{J}$ of $Sk(X)$, such that there is a unique $1$-dimensional face of $\Delta_{\mathcal{X}}$ containing $\Delta_{J}$, and it lies on $Sk(X)$. The simplest possibility $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})=0$ imposes a Neumann-like boundary condition on $\phi_{0}$. This example may describe the Tian-Yau region in [25][42].

Example 5.4. For $m=1$, consider a $0$-dimensional face $\Delta_{J}$ of $Sk(X)$, such that there are two $1$-dimensional faces of $\Delta_{\mathcal{X}}$ containing $\Delta_{J}$, and they both lie on $Sk(X)$. A very simple geometric situation is when $E_{J}$ is the smooth total space of a possibly singular $\mathbb{P}^{1}$-fibration over an $(n-1)$-dim smooth variety $D$, and the two 1-dimensional faces correspond to two disjoint sections $E_{J^{\prime},1},E_{J^{\prime},2}$ of the $\mathbb{P}^{1}$-fibration, so $E_{J^{\prime},1}\simeq E_{J^{\prime},2}\simeq D$. A natural way to make $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n}\cdot E_{J}=0$ and $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})$ nef for $x\in\Delta_{J}$, is to ask $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})$ to be the pullback of a nef class on the base $D$. We regard this nef class as the limiting element of $\mathcal{D}_{J}(y,\left\lVert\cdot\right\rVert_{CY})\in H^{1,1}(E_{J^{\prime},i})$ as $y$ approaches $x$ from either of the $1$-dimensional faces. Then the matching condition is naturally seen as the continuity of $\mathcal{D}_{J}(y,\left\lVert\cdot\right\rVert_{CY})$ across the $0$-dimensional face. This example may be relevant for the Ooguri-Vafa type neck region in [25][42] (cf. [42, section 7.1]). The possibility for the $\mathbb{P}^{1}$-fibration to develop nodal fibres is related to the monopole bubbling phenemenon in these papers.