Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons

Let (X,L) be a (semi-) polarized complex projective variety and T a real torus acting holomorphically on X with moment polytope P. Given a probability density g on P we introduce a new type of Monge-Ampere measure on X, defined for singular T-invariant metrics on the line bundle L, generalizing the ordinary Monge-Ampere of global pluripotential theory, which corresponds to the case when T is trivial (or g=1). In the opposite extreme case when T has maximal rank, i.e. (X,L,T) is a toric variety, the solution of the corresponding Monge-Ampere equation with right hand side \mu corresponds to the convex Kantorovich potential for the optimal transport map in the Monge-Kantorovich transport problem betweeen \mu and g (for a quadratic cost function). Accordingly, our general setting can be seen as a complex version of optimal transport theory. Our main complex geometric applications concern the pluripotential study of singular (shrinking) Kahler-Ricci solitons. In particular, we establish the uniqueness of such solitons, modulo automorphisms, and explore their relation to a notion of modified K-stability inspired by the work of Tian-Zhu. The quantization of this setup, in the sense of Donaldson, is also studied.