论文标题
在代数锥平滑上的单数拉格朗日圆环纤维
Singular Lagrangian torus fibrations on the smoothing of algebraic cones
论文作者
论文摘要
给定一个晶格polytope $ q \ subset \ mathbb {r}^n $，我们可以考虑锥体$σ= c（q）= \ {λ（q，1）\ in \ mathbb {r}^{r}^{n+1} |λ} |λ\ in \ in \ mathbb {r} \ Mathbb {r}^{n+1} $，以及与$σ$相关的仿射感谢物$y_σ$。阿尔特曼（Altmann）表明，$y_σ$的价值变形空间可以用polytope $ q $的Minkowski分解来描述。在$ Q $的某些条件下，我们可以使用Altmann的结果获得$y_σ$的平滑变形$ y__ $。在本文中，我们考虑了一些$ \ mathbb {c}^n $内部的$y_ε$，并在$y_ε$上构建复杂的纤维，其中一般纤维$（\ mathbb {c}^*）^n $和使用与Minkowski Decompstorents的组件相关的全球坐标所描述的有限单数纤维。我们从复杂的纤维中构造了一个奇异的Lagrangian圆环。这种奇异振动允许凸基图表示，切成剪切，作为Symington所描述的几乎要复曲面纤维的基本图的自然概括（$ \ dim = 4 $）。特别是，我们获得了一个凸基图，其图像是$ c（q）$的双锥。这些纤维中的每一个中都有一个1参数单调拉格朗日摩托车家族。使用隔离式配方，我们用$ Q $的Minkowski分解来描述与该家族相关的潜力，从而恢复了LAU的结果，并讨论了非替代性。我们还讨论了结果的其他后果。
Given a lattice polytope $Q\subset \mathbb{R}^n$, we can consider the cone $σ=C(Q)=\{λ(q,1)\in \mathbb{R}^{n+1}|λ\in \mathbb{R}_{\geq0}, q\in Q\} \subset \mathbb{R}^{n+1}$, and the affine toric variety $Y_σ$ associated to $σ$. Altmann showed that the versal deformation space of $Y_σ$ can be described by the Minkowski decomposition of the polytope $Q$. Under some conditions on $Q$, we can obtain a smooth deformation $Y_ε$ of $Y_σ$ using Altmann's result. In this article, we consider $Y_ε$ inside some $\mathbb{C}^N$ and construct a complex fibration on $Y_ε$, with general fibre $(\mathbb{C}^*)^n$ and finite singular fibres described using global coordinates related to the components of the Minkowski decomposition. We construct a singular Lagrangian torus fibration out of the complex fibration. This singular fibration admits a convex base diagram representation with cuts as a natural generalization of base diagrams described by Symington for Almost Toric Fibrations ($\dim=4$). In particular, we obtain a convex base diagram whose image is the dual cone of $C(Q)$. There is a 1-parameter family of monotone Lagrangian tori in each of these fibrations. Using the wall-crossing formula, we describe the potential associated with this family in terms of the Minkowski decomposition of $Q$, recovering the result of Lau, and discuss non-displaceability. We also discuss some other consequences of our results.
