The cutting construction of toric symplectic and contact manifolds

TL;DR

This paper introduces a cutting construction method for non-compact symplectic toric manifolds and contact manifolds, expanding the toolkit beyond Delzant's construction and analyzing their Sasakian structures.

Contribution

The paper develops a new cutting construction for toric symplectic and contact manifolds, especially those associated with weakly convex cones, and investigates their Sasakian properties.

Findings

Constructed toric symplectic cones from weakly convex good cones.

Showed the non-existence of toric Sasakian structures on these manifolds.

Established that contact toric manifolds of toric K-contact type are of toric Sasakian type.

Abstract

We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to a weakly convex good cone by the cutting construction. (Note that these toric contact manifolds can not be constructed by Delzant construction.) We further prove there are no toric Sasakian structures on these contact manifolds. From this, contact toric manifolds of toric K-contact type are of toric Sasakian type.