A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry

By B. Berndtsson. 2013

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For $$\phi $$ϕ a metric on the anticanonical bundle, $$-K_X$$-KX, of a Fano manifold $$X$$X we consider the volume of $$X$$X$$\begin{aligned} \int _X e^{-\phi }. \end{aligned}$$∫Xe-ϕ.In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on $$-K_X$$-KX. Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $$X$$X, even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando–Mabuchi uniqueness theorem for Kähler–Einstein metrics. A generalization of this theorem to ‘twisted’ Kähler–Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than $$-K_X$$-KX, and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kähler–Ricci solitons.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/04c740c011c815bc2609054a0e58ea9d2d9ee953

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Citations in the corpus

Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, background): “ V deﬁned on Xreg,which, by normality, forces f= 0,i.e. F∗ tV = V,as desired. To see that there exists a vector ﬁeld W as in the theorem, we recall that the proof of the convexity properties of DV in [15, 14] realizes Ft by integrating a family of holomorphic vector ﬁelds Vt where Vt has Hamiltonian”
Integral bounds on curvature and Gromov–Hausdorff limits (2012, methodology): “on 3. Let Aut(Z) be the group of holomorphic transformations of Zthat preserves ξ0; in the notation of Section 3, this is a subgroup of Gξ0 that ﬁxes [Z] in Hilb. The following results were proved in [4] and [9] for Kahler-Einstein Q-Fano varieties. 35 Proposition 4.8 (Bando-Mabuchi theorem). Ricci-”
Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, background): “Injectivity follows by BandoMabuchi uniqueness and its generalization to the singular setting [7].”
Kahler-Einstein metrics with edge singularities (2011, background): “When μ > 0, the uniqueness of the (twisted) Ricci iteration was proven recently by Berndtsson [12], and it follows from his result that whenever the twisted K-energy E 0 is proper, then also the Kähler–Einstein edge metric must be unique.”
Regularity of Kähler–Ricci flows on Fano manifolds (2013, methodology): “IfM admits a shrinking Kähler–Ricci soliton, then by the uniqueness theorem of Berndtsson [4] and Berman–Boucksom–Essydieux– Guedj–Zeriahi [3], the Kähler–Ricci flow should converge to the Ricci soliton.”
On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations (2020, background): “By [14], LΘ (or D since Eg is affine) is convex along a geodesic.”
Kähler–Einstein metrics along the smooth continuity method (2015, methodology): “As argued in [11], it can be shown that there always exists a smooth solution νs to (105) satisfying ∂̄νs ∧ ω = 0.”
The Kähler–Ricci flow and optimal degenerations (2016, background): “By Berndtsson [6], the Ding functional is convex along geodesics, and so Y (t) is monotonically decreasing in t.”
Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II (2015, methodology): “9 can be proved using arguments analogous to the appendix of [9], with the main technical input from [4] and [3].”