A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem

By B. Berndtsson. 2011

67 citations recorded by the source corpus.

For $\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-\phi}. $$ We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$. As consequences we get a simplified proof of the Bando-Mabuchi uniqueness theorem for K\"ahler - Einstein metrics and a generalization of this theorem to 'twisted' K\"ahler-Einstein metrics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5d7b22150e42fbf0f4c91148788bdb641e9af2e8

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, methodology): “Berndtsson’s result was applied in the current set-up in the work of [10] and [8, Lemma 6.”
Kahler-Einstein metrics with edge singularities (2011, methodology): “conclude that {ψ˜kτ} subconverges to the potential of a Ka¨hler–Einstein edge metric. Whenever it is unique, the iteration itself necessarily converges. Berndtsson’s generalized Bando–Mabuchi Theorem [4, 9] shows uniqueness of Ka¨hler–Einstein edge metrics up to an automorphism (which must be tangen”
Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2\pi\ and completion of the main proof (2013, methodology): “he proof of Theorem 1. 6 Appendix 1: Convexity of Ding functional and uniqueness of weak conical K¨ahler-Einstein metrics We reproduce the proof of Proposition 14 and Proposition 15, mainly following [6] (compare also [5]). A more detailed account of these can be found in the expository note by Long”
Geometric pluripotential theory on Kähler manifolds (2019, result): “Inspired from this, a careful analysis of infinite dimensional spaces led Bando–Mabuchi to prove uniqueness of KE metrics [2], and later Berman–Berndtsson to discover even more general uniqueness results [8].”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, background): “hler-Einstein metrics with edge-cone singularities. It is interesting to compare the uniqueness property for toric KählerEinstein metrics contained in the previous theorem with the general results in [14, 12] saying that any two log Kähler-Einstein metrics associated to a given log Fano variety (X,∆”
K‐Stability and Kähler‐Einstein Metrics (2012, background): “k/jM1 for ` D k`, so we get a holomorphic section 1 29 In [5], based on arguments from [6], Berman-Boucksom-Essydieux-Guedj-Zeriahi proved a strong uniqueness theorem: Kähler-Einstein metrics on (possibly) singular Fano varieties are unique modulo an automorphism group obtained by complexifying an i”
Existence and deformations of Kähler–Einstein metrics on smoothable $\mathbb{Q}$-Fano varieties (2014, background): “Φǫ t) and gǫ(t) = F1 ωt (Φǫ t) Then by deﬁnition the log-Ding-functional Fω t,(1−β)Dt(Φ ǫ t) is the sum of these two functions. By the general theory on positivity of direct image bundle according to [8] and positivity of Deligne Pairing according to [46], both pieces of the log-Ding-functional are ”
Ricci flow on surfaces with conic singularities (2013, result): “1−βi)pi) is nonnegative, or equivalently, that this pair is logarithmically K-stable. The generalization of the uniqueness part of this result to higher dimensions has been accomplished by Berndtsson [4]. Non-existence when coercivity fails can be easily deduced from [20], together with work of Berm”
The Mabuchi Geometry of Finite Energy Classes (2014, unknown): “ity in the t−variable, we can deﬁne v˙0 = limt→0(v t−v0)/tand v˙ j 0 = limt→0(v j−v 0)/t. The following holds: lim j→∞ kv˙j 0kχ,v0 = kv˙0kχ,v0. Proof. By an observation of Berndtsson (see Section 2.1 [Brn1]), there exists C >0 such that kv˙0kL∞(X),kv˙ j 0kL∞(X) ≤ C. We also have v≤ v j, j∈ N by t”