Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons

Abstract

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 μcorresponds to the convex Kantorovich potential for the optimal transport map in the Monge-Kantorovich transport problem betweeen μand 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.

41 pages

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Cited by in corpus (25)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• Sasaki-Einstein metrics and K-stability
• K-Stability of Fano spherical varieties
• Kähler metrics with constant weighted scalar curvature and weighted K-stability
• Classification results for expanding and shrinking gradient Kähler-Ricci solitons
• Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties
• Kähler-Einstein metrics along the smooth continuity method
• The existence of the Kähler-Ricci soliton degeneration
• The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics
• Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics
• Basis divisors and balanced metrics
• Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons
• Conformally Kähler, Einstein--Maxwell metrics and boundedness of the modified Mabuchi-functional
• Stability of anti-canonically balanced metrics
• Modified Futaki invariant and equivariant Riemann-Roch formula
• The moduli space of Fano manifolds with Kähler-Ricci solitons
• Fano-Ricci limit spaces and spectral convergence
• Steady Kähler-Ricci solitons on crepant resolutions of finite quotients of ${\mathbb{C}}^n$
• On the modified Futaki invariant of complete intersections in projective spaces
• Existence of Kahler-Ricci solitons on smoothable Q-Fano varities
• Smooth approximation of the modified conical Kähler-Ricci flow
• Optimal transport on completely integrable toric manifolds
• The Kähler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations
• K-stability and polystable degenerations of polarized spherical varieties
• Kähler-Ricci flow for deformed complex structures