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

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 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 Kähler - Einstein metrics and a generalization of this theorem to 'twisted' Kähler-Einstein metrics.

22 pages, revised and expanded

References in corpus (3)

• A variational approach to complex Monge-Ampere equations
• Analytic torsion, vortices and positive Ricci curvature
• Regularity of plurisubharmonic upper envelopes in big cohomology classes

Cited by in corpus (13)

• K-stability and Kähler-Einstein metrics
• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• Conical Kahler-Einstein metric revisited
• Kahler-Einstein metrics and stability
• Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
• Smooth and singular Kahler-Einstein metrics
• On the existence of conic Kahler-Einstein metrics
• On the limit of spectral measures associated to a test configuration
• Properness of log $F$-functionals
• A Bando-Mabuchi Uniqueness Theorem
• Stability of the conical Kähler-Ricci flows on Fano manifolds
• Isometry group of Sasaki-Einstein metric