Delzant-type classification of near-symplectic toric 4-manifolds

Abstract

Delzant's theorem for symplectic toric manifolds says that there is a one-to-one correspondence between certain convex polytopes in $\mathbb{R}^n$ and symplectic toric $2n$-manifolds, realized by the image of the moment map. I review proofs of this theorem and the convexity theorem of Atiyah-Guillemin-Sternberg on which it relies. Then, I describe Honda's results on the local structure of near-symplectic 4-manifolds, and inspired by recent work of Gay-Symington, I describe a generalization of Delzant's theorem to near-symplectic toric 4-manifolds. One interesting feature of the generalization is the failure of convexity, which I discuss in detail. The first three chapters are primarily expository, duplicate material found elsewhere, and may be skipped by anyone familiar with the material, but are included for completeness.