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

arXiv1103.0923MaRDI QIDQ6223942

Bo Berndtsson

Publication date: 4 March 2011

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