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

By Robert J. Berman, David Witt Nyström. 2014

87 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3fb2996fc00f192730ac433477b9fef59b7b9e4b

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Citations in the corpus

Kähler–Ricci flow, Kähler–Einstein metric, and K–stability (2015, methodology): “ected components. Thus we only need to show that for a given T′, and a connected component Hilb] of T′, for all Z ∈ C with T(Z) = T′ and [Z] ∈ Hilb], JVZ gives rise to the same element in Lie(T′). By [1] we know for given Z, VZ is characterized as the unique vector in Rk that satisﬁes FutJVZ (V ′) =”
K-stability of Fano spherical varieties (2016, methodology): “ We call Hamiltonian function of ξ∈awith respect to ωthe smooth function θξ on Xreg deﬁned by Lξω= i∂∂θ¯ ξ, where Lξ is the Lie derivative with respect to ξ. It is shown to exist and to be bounded in [BW]. This function is well deﬁned up to a constant and we ﬁx this constant by the normalization con”
The Kähler–Ricci flow and optimal degenerations (2016, background): “Indeed, both quantities are limit derivatives of a component of the Ding functional [4, 3].”
On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations (2020, methodology): “hen Liu forhelpful discussions about equivariant K-stability. We would like to thank Bin Zhou for helpful comments. 2. Functionals and their dualities In this section, we will ﬁrst review a result in [13] which deﬁnes the Radon measure gϕ(ddcϕ)n for all T-invariant L-psh Hermitian metrics. We will g”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, background): “φ)ω [n] φ . where (s,s′)kφ := hk φ(s,s ′). The operators (A(k) ξj )j=1,···,ℓ are Hermitian with respect to h·,·iv,kφ, with spectrum contained in the momentum polytope P (see lemma 13). Deﬁnition A.1. [12, 80, 91] Let φ∈ KT ω, {si | i= 0,··· ,Nk} be a h·,·iv,kφ-orthonormal basis of H and w ∈ C∞(P,R).”
Classification results for expanding and shrinking gradient Kähler–Ricci solitons (2019, methodology): “ on shrinking gradient Ka¨hler-Ricci solitons presented here allow us to speculate on possible deeper connections between such metrics and algebraic geometry. In the compact case, Berman-Witt-Nystrom [BN14] gave an algebraic formula for the weighted volume functional and its derivative. We generaliz”