Kähler-Einstein metrics on toric Fano manifolds and connections to Optimal Transport

Abstract

Abstract We introduce the problem of finding a Kähler-Einstein metric on a Kähler manifold and specifically on a Fano manifold. We restrict to the class of toric complex manifolds where the symmetry can be used to reduce the resulting partial differential equation to a real equation in Rn. We then introduce the theory of optimal transport, specially adapted to the application. We present a special transportation problem which is in fact equivalent to a weak formulation of the Kähler-Einstein equation on toric Fano manifolds. The presentation is a literature study aimed at presenting the material in a self-contained and elementary fashion. We also present a novel variational approach to the existence problem in the language of optimal transport and equilibrium physics. We show some results towards an existence result based on this approach. Finally we exemplify large parts of the theory on complex projective space, an explicit example of a toric Fano manifold. We also compute the free energy, an invariant we will introduce, on complex projective space.