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

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

2011-03-04

arxiv.org/abs/1103.0923v3

Mathematics

Differential Geometry

22 pages, revised and expanded

Scientific paper

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.

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