Hamiltonian torus actions on symplectic orbifolds and toric varieties

TL;DR

This paper extends fundamental theorems to symplectic orbifolds and classifies certain orbifolds with torus actions as algebraic toric varieties, linking symplectic geometry with algebraic geometry.

Contribution

It proves orbifold versions of the abelian connectedness and convexity theorems and classifies compact symplectic orbifolds with integrable torus actions.

Findings

Orbifold versions of abelian connectedness and convexity theorems established.

Classification of symplectic orbifolds with torus actions via convex rational polytopes.

All such orbifolds are shown to be algebraic toric varieties.

Abstract

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each facet and that all such orbifolds are algebraic toric varieties.