Singular Lagrangian torus fibrations on the smoothing of algebraic cones

Abstract

Given a lattice polytope $Q\subset {\mathbb{R}}^n$, we can consider the cone $\sigma =C(Q)=\{\lambda (q,1)\in {\mathbb{R}}^{n+1}|\lambda \in {\mathbb{R}}_{\ge 0},q\in Q\}\subset {\mathbb{R}}^{n+1}$, and the affine toric variety $Y_{\sigma }$ associated to $\sigma$. Altmann showed that the versal deformation space of $Y_{\sigma }$ can be described by the Minkowski decomposition of the polytope $Q$. Under some conditions on $Q$, we can obtain a smooth deformation $Y_{\epsilon }$ of $Y_{\sigma }$ using Altmann's result. In this article, we consider $Y_{\epsilon }$ inside some ${\mathbb{C}}^N$ and construct a complex fibration on $Y_{\epsilon }$, with general fibre $({\mathbb{C}}^{\ast }{)}^n$ 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.

43 pages, 14 figures. Accepted version