Symplectic fillability of toric contact manifolds

TL;DR

This paper proves that higher-dimensional compact toric contact manifolds are generally symplectically fillable, contrasting with the non-fillability of certain 3-dimensional cases, based on classification and structural analysis.

Contribution

It establishes the fillability of all higher-dimensional compact toric contact manifolds, expanding understanding beyond the known 3-dimensional non-fillable cases.

Findings

Higher-dimensional toric contact manifolds are weakly fillable.

Most higher-dimensional cases are strongly fillable.

Certain 3D toric contact manifolds are non-fillable due to overtwistedness.

Abstract

According to Lerman, compact connected toric contact 3-manifolds with a non-free toric action whose moment cone spans an angle greater than $\pi$ are overtwisted, thus non-fillable. In contrast, we show that all compact connected toric contact manifolds in dimension greater than three are weakly symplectically fillable and most of them are strongly symplectically fillable. The proof is based on the Lerman's classification of toric contact manifolds and on our observation that the only contact manifolds in higher dimensions that admit free toric action are the cosphere bundle of $T^d, d\geq3$ $(T^d\times S^{d-1})$ and $T^2\times L_k,$ $k\in\mathbb{N},$ with the unique contact structure.