ScalingStacks

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.