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

TL;DR

This paper proves a concavity property of the volume functional on Fano manifolds, leading to a simplified proof and generalization of the Bando-Mabuchi uniqueness theorem for Kähler-Einstein metrics.

Contribution

It establishes a Brunn-Minkowski type inequality for Fano manifolds and extends the Bando-Mabuchi theorem to twisted Kähler-Einstein metrics.

Findings

Logarithm of volume is concave along geodesics in the space of metrics.

Concavity is strict unless the geodesic is generated by a holomorphic vector field.

Provides a simplified proof and a generalization of the Bando-Mabuchi theorem.

Abstract

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.