On complex and symplectic toric stacks

TL;DR

This paper explores the relationship between symplectic and algebraic constructions of toric stacks, showing their equivalence in certain cases and serving as an introductory guide for algebraically inclined readers.

Contribution

It demonstrates the isomorphism between stacks from Lerman-Tolman and Borisov et al. constructions when induced by a polytope, bridging symplectic and algebraic perspectives.

Findings

Stacks from Lerman-Tolman are isomorphic to Borisov et al. stacks for polytope-induced stacky fans.

Provides an example-driven introduction to symplectic toric geometry.

Clarifies the relationship between different constructions of toric stacks.

Abstract

Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant. In algebraic geometry, there is a more general construction using fans rather than polytopes. However, in case the fan is induced by a smooth polytope Audin showed both constructions to give isomorphic projective varieties. For rational but not necessarily smooth polytopes the Delzant construction was refined by Lerman and Tolman, leading to symplectic toric orbifolds or more generally, symplectic toric DM stacks (Lerman and Malkin). We show that the stacks resulting from the Lerman-Tolman construction are isomorphic to the stacks obtained by Borisov et al. in case the stacky fan is induced by a polytope. No originality is claimed (cf. also an article by Sakai). Rather we hope that this text serves as an example driven introduction to symplectic toric geometry for the algebraically minded reader.