Concave symplectic toric fillings

TL;DR

This paper demonstrates that every contact toric 3-manifold has infinitely many distinct concave symplectic toric fillings, constructed via specific plumbings, refining previous results in the non-toric setting.

Contribution

It extends prior work by showing the existence of infinitely many non-equivariantly symplectomorphic toric fillings for contact toric 3-manifolds.

Findings

Existence of infinitely many non-equivariantly symplectomorphic toric fillings

Construction of these fillings via linear and cyclic plumbings over spheres

Refinement of previous non-toric results to the toric setting

Abstract

As shown by Etnyre and Honda in [EH], every contact 3-manifold admits infinitely many concave symplectic fillings that are mutually not symplectomorphic and not related by blow ups. In this note we refine this result in the toric setting by showing that every contact toric 3-manifold admits infinitely many concave symplectic toric fillings that are mutually not equivariantly symplectomorphic and not related by blow ups. The concave symplectic toric structure is constructed on certain linear and cyclic plumbings over spheres.