SYZ and optimal transport stability of Weyl polytopes

TL;DR

This paper establishes optimal transport stability for reflexive Weyl polytopes and demonstrates the validity of the weak metric SYZ conjecture for certain Fano toric manifolds, linking geometric stability with mirror symmetry.

Contribution

It proves optimal transport stability for reflexive Weyl polytopes and confirms the weak metric SYZ conjecture for centrally symmetric smooth Fano toric manifolds.

Findings

Optimal transport stability for reflexive Weyl polytopes.

Validation of the weak metric SYZ conjecture for specific Fano toric manifolds.

Connection between Weyl polytope stability and mirror symmetry conjectures.

Abstract

We prove optimal transport stability (in the sense of Andreasson and the second author) for reflexive Weyl polytopes: reflexive polytopes which are convex hulls of an orbit of a Weyl group. When the reflexive Weyl polytope is Delzant, it follows from work of Li, Andreasson, Hultgren, Jonsson, Mazzon, McCleerey, that the weak metric SYZ conjecture holds for the Dwork family in the corresponding toric Fano manifold. In particular, we show that the weak metric SYZ conjecture holds for centrally symmetric smooth Fano toric manifolds.