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

Journal article, 2015

For ϕ a metric on the anticanonical bundle, −KX , of a Fano manifold X we consider the volume of X ∫Xe−ϕ. In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on −KX . 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 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 −KX , and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kähler–Ricci solitons.