Emmy-Noether-Seminar

How to distinguish between symplectic toric manifolds?

Milena Pabiniak (Köln)

Abstract: The heroes in this talk are symplectic toric manifolds, i.e. manifolds equipped with a closed, non-degenerate 2-form, called a symplectic structure, and many symmetries, coming from an action of a torus of big dimension. By a result of Delzant they are classified by convex polytopes. This correspondence builds a bridge between symplectic geometry and combinatorics of convex polytopes and led to proving many important results on both sides of the bridge.

Delzant's classification is a classification up to equivariant symplectomorphisms, i.e. diffeomorphisms respecting not only the symplectic form but also the torus action. A classification of symplectic toric manifolds up to diffeomorphisms or symplectomorphisms remains mysterious. We still cannot recognize when two different polytopes correspond to the same symplectic manifold equipped with two different toric actions. Masuda and Suh suggested that maybe the integral cohomology ring could classify symplectic toric manifolds up to diffeomorphisms.

In this talk I will introduce symplectic toric manifolds, discuss the well-understood Delzant classification via polytopes and show how to calculate the integral cohomology ring of a symplectic toric manifold from the associated polytope.

Then I will pose and partially answer the following natural question: is any ring isomorphism F: H*(M;Z) -> H*(N;Z), which maps the class of a symplectic form of M to the class of a symplectic form of N, induced by a symplectomorphism f: N -> M?

I will explain how one can use a construction from algebraic geometry, called toric degeneration, to construct required symplectomorphisms and how such non-equivariant symplectomorphisms affect polytopes associated to the symplectic toric manifolds in question. There will be lots of pictures.

The talk is based on a joint work with Sue Tolman.