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

Abstract

For $\varphi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $${\int }_X{e}^{-\varphi }.$$ We prove that the logarithm of the volume is concave along bounded 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 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 $-K_X$, and finally use the same method to give a new proof of the theorem of Tian and Zhu of uniqueness of Kähler-Ricci solitons. This is an expanded version of an earlier preprint, "A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem", arXiv:1103.0923

This is a revised and expanded version of ArXiv 1103.0923

References in corpus (3)

• A variational approach to complex Monge-Ampere equations
• Regularity of plurisubharmonic upper envelopes in big cohomology classes
• Strict and non strict positivity of direct image bundles

Cited by in corpus (6)

• Ricci flow on surfaces with conic singularities
• Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons
• Moment maps, nonlinear PDE, and stability in mirror symmetry
• Connecting toric manifolds by conical Kahler-Einstein metrics
• Regularity of Kähler-Ricci flows on Fano manifolds
• Twisted and conical Kähler-Ricci soliton on Fano manifolds