On toric degenerations and cohomological rigidity for symplectic toric manifolds

Given an integral polytope $\Delta \subset \mathbb{R}^n$, there is a naturally associated toric variety $ X_\Delta$, that is, an $n$-dimensional algebraic variety with an $n$-dimensional torus action. We say that $X_\Delta$ is a toric manifold if the variety is smooth. In this case, $X_\Delta$ inherits a symplectic form $\omega_\Delta \in \Omega^2(X_\Delta)$. A conjecture, called “cohomological rigidity”, posits that any toric manifolds with isomorphic cohomology rings are diffeomorphic. This has been proved in various special cases. There's a natural symplectic analog of this conjecture: If there is an isomorphism of cohomology rings which preserves the symplectic cohomology class, then the manifolds are symplectomorphic. Unfortunately, this has been difficult to prove, because it is hard to construct symplectomorphisms. Adapting ideas from Harada and Kaveh, we use toric degenerations to find symplectomorphisms between certain toric manifolds. This generalizes the well-known isomorphisms between different Hirzebruch surfaces. We then use this to prove the symplectic analog of cohomological rigidity under certain assumptions. For example, it holds when the cohomology ring is isomorphic to the cohomology ring of the product of $n$ two-spheres. This talk is based on work-in-progress, which is joint with Milena Pabiniak.