On complex and symplectic 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 [D]. 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 [Au] showed both constructions to give isomorphic projective varieties. For rational but not necessarily smooth polytopes the Delzant construction was refined by Lerman and Tolman [LT], leading to symplectic toric orbifolds or more generally, symplectic toric DM stacks [LM]. We show that the stacks resulting from the Lerman–Tolman construction are isomorphic to the stacks obtained by Borisov et al. [BCS] in case the stacky fan is induced by a polytope. No originality is claimed (cf. also the article by Sakai [S]). Rather we hope that this text serves as an example driven introduction to symplectic toric geometry for the algebraically minded reader.