SYZ and optimal transport stability of Weyl polytopes

TL;DR

The paper establishes optimal transport stability for reflexive Weyl polytopes by exploiting symmetry under the Weyl group and dual reflections, and uses this to prove the weak metric SYZ conjecture for certain toric Fano hypersurfaces, notably the Dwork family in centrally symmetric manifolds. The approach relies on a key observation that invariant OT plans are supported on unions of positive half-spaces, which, together with Weyl symmetry, yields explicit support descriptions and Monge–Ampère equation solutions on the boundary. When the polytopes or their duals satisfy the Delzant condition, the weak SYZ conjecture holds; higher regularity results follow by reducing to planar OT subproblems and obtaining $C^{1,\,\alpha}$ bounds away from lower-dimensional discriminants. The paper also classifies three-dimensional reflexive polytopes in terms of Weyl structure, identifying which polytopes fail the vertex condition and compiling infinite families of Weyl polytopes, thereby connecting combinatorial geometry with toric Calabi–Yau–type degenerations.

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.

Figures (1)

• Figure 1: Weyl polytopes for the root system $B_2$

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Proposition 2.6
• Definition 2.7
• Proposition 2.8