Singular Lagrangian torus fibrations on the smoothing of algebraic cones

TL;DR

This paper constructs singular Lagrangian torus fibrations on smoothings of algebraic cones derived from lattice polytopes, linking the geometry to Minkowski decompositions and potential functions, with implications for symplectic topology.

Contribution

It introduces a method to build singular Lagrangian torus fibrations on smoothings of algebraic cones, generalizing base diagrams and connecting to Minkowski decompositions.

Findings

Constructed complex fibrations with singular Lagrangian torus fibers.

Described the potential functions using Minkowski decompositions.

Established non-displaceability of certain Lagrangian tori.

Abstract

Given a lattice polytope $Q\subset \mathbb{R}^n$, we can consider the cone $\sigma=C(Q)=\{\lambda(q,1)\in \mathbb{R}^{n+1}|\lambda \in \mathbb{R}_{\geq0}, q\in Q\} \subset \mathbb{R}^{n+1}$, 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 $(\mathbb{C}^*)^n$ 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.