Abstract

I will discuss some joint work with Julius Ross.

In toric geometry, line bundles are associated with polytopes. In 1996 Andrei Okounkov found a way to generalize this, so that any ample line bundle L gets an associated convex body, called the Okounkov body.

However, while the toric construction encodes the symplectic geometry of the variety, Okounkov’s construction is of a purely algebro-geometric nature. We wonder if there is a corresponding symplectic interpretation of the Okounkov body, involving the symplectic form defined by the curvature form of a fixed metric on L, as there is in the toric case?

By setting up a certain homogeneous Monge-Ampère equation, we show that we can accomplish this, given some regularity assumptions on the solutions to the HMAE . In one dimension the problem is equivalent to finding a solution to the Hele-Shaw flow. Recall that this flow describes the propagation of a fluid being injected in between two plates that are close to each other.