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

Abstract

For \(\phi \) a metric on the anticanonical bundle, \(-K_X\), of a Fano manifold \(X\) we consider the volume of \(X\) $$\begin{aligned} \int _X e^{-\phi }. \end{aligned}$$ In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on \(-K_X\). Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on \(X\), even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando–Mabuchi uniqueness theorem for Kahler–Einstein metrics. A generalization of this theorem to ‘twisted’ Kahler–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 on uniqueness of Kahler–Ricci solitons.