Complex geometry and optimal transport

Optimal transport studies the most economical movement of resources. In other words, one considers a pile of raw material and wants to transport it to a final configuration in a cost-efficient way. Under quite general assumptions, the solution to this problem will be induced by a transport map where the mass at each point in the initial distribution is sent to a unique point in the target distribution. In this talk, we will discuss the regularity of this transport map (i.e., whether nearby points in the first pile are sent to nearby points in the second pile).

It turns out there are both local and global obstructions to establishing smoothness for the transport. When the cost is induced by a convex potential, we show that the local obstruction corresponds to the curvature of an associated Kahler manifold and discuss the geometry of this curvature tensor. In particular, we show that its negativity is preserved along Kahler-Ricci flow.