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

Abstract

For $ϕ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-ϕ}. $$ 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.