[Paper Review] Symplectic toric orbifolds

Eugene Lerman, Susan Tolman|ArXiv.org|Dec 23, 1994

Geometric and Algebraic Topology6 references9 citations

TL;DR

This paper establishes a one-to-one correspondence between symplectic toric orbifolds and convex rational simple polytopes equipped with positive integers on each facet. By generalizing Delzant's theorem to orbifolds, the authors show that the classification of such symplectic manifolds reduces to combinatorial data, extending toric geometry to singular spaces with isotropy.

ABSTRACT

A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic form $ω$ on $M$, and an effective Hamiltonian action of a torus $T$ on $M$, where the dimension of $T$ is half the dimension of $M$. We prove that there is a one-to-one correspondence between symplectic toric orbifolds and convex rational simple polytopes with positive integers attached to each facet.

Motivation & Objective

• To extend the classification of symplectic toric manifolds to the case of orbifolds with singularities.
• To identify the necessary and sufficient combinatorial data that classify symplectic toric orbifolds.
• To generalize Delzant's theorem from smooth manifolds to orbifolds by incorporating isotropy data.
• To establish a precise correspondence between geometric structures and convex rational simple polytopes with integer weights on facets.
• To provide a framework for understanding Hamiltonian torus actions on orbifolds via convex geometry.

Proposed method

• Define symplectic toric orbifolds as compact connected orbifolds with a Hamiltonian effective torus action of half the dimension.
• Introduce a moment map that maps the orbifold to a convex rational simple polytope in the dual of the Lie algebra of the torus.
• Attach a positive integer to each facet of the polytope, representing the order of the isotropy group at the preimage of the facet.
• Use the combinatorics of the polytope and the facet weights to reconstruct the orbifold structure and symplectic form.
• Prove that the assignment of weights to facets is invariant under symplectomorphisms preserving the torus action.
• Demonstrate that every such polytope with positive integers on facets arises from a unique symplectic toric orbifold up to equivariant symplectomorphism.

Experimental results

Research questions

• RQ1 What combinatorial data classify symplectic toric orbifolds under equivariant symplectomorphism?
• RQ2 How does the presence of orbifold singularities affect the moment map image and its structure?
• RQ3 Can Delzant's classification theorem for symplectic toric manifolds be extended to orbifolds?
• RQ4 What is the role of integer weights on polytope facets in encoding isotropy data of the orbifold?
• RQ5 Is the correspondence between symplectic toric orbifolds and weighted rational polytopes bijective and functorial?

Key findings

• There exists a one-to-one correspondence between isomorphism classes of symplectic toric orbifolds and convex rational simple polytopes with positive integers assigned to each facet.
• The moment map image of a symplectic toric orbifold is a convex rational simple polytope, and the isotropy data is encoded in the facet weights.
• The integer attached to each facet corresponds to the order of the isotropy group of the corresponding characteristic suborbifold.
• The classification is invariant under equivariant symplectomorphisms, ensuring uniqueness of the orbifold structure from the combinatorial data.
• The construction generalizes Delzant's theorem from smooth manifolds to orbifolds, providing a complete classification in the singular setting.
• The symplectic form and torus action are completely determined by the polytope and its facet weights, up to equivariant symplectomorphism.