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 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 $-K_X$, and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kähler–Ricci solitons.

Appendix

Here we will prove Proposition 10.1. We suppress the dependence of $\phi$ on $t$ and $s$ in the formulas and use subscripts only to denote differentiation with respect to these variables.

$$\begin{aligned} \frac{d}{ds}\int \dot{\phi _t}(i\partial \bar{\partial }\phi )^n e^{a h^{\phi }}&= \int \ddot{\phi }_{t,s} (i\partial \bar{\partial }\phi )^n e^{a h^{\phi }} \\&\quad +\,n\int \dot{\phi _t}(i\partial \bar{\partial }\dot{\phi }_s)\wedge (i\partial \bar{\partial }\phi )^{n-1} e^{a h^{\phi }}\\&\quad +\,a\int \dot{\phi _t}V(\dot{\phi }_s)\wedge (i\partial \bar{\partial }\phi )^{n} e^{a h^{\phi }}=:I+II+III. \end{aligned}$$

Integrating by parts we get

$$\begin{aligned} II&= - n\int i\partial \dot{\phi _t}\wedge \bar{\partial }\dot{\phi _s}\wedge (i\partial \bar{\partial }\phi )^{n-1} e^{ah^\phi }\\&-\,an\int i\dot{\phi _t}\partial h^{\phi }\wedge \bar{\partial }\dot{\phi _s}\wedge (i\partial \bar{\partial }\phi )^{n-1} e^{ah^\phi }. \end{aligned}$$

Recall that $i\bar{\partial }h^\phi =V\rfloor i\partial \bar{\partial }\phi$ so that we have $-i\partial h^\phi =\bar{V}\rfloor i\partial \bar{\partial }\phi .$ Since contraction with a vector field is an antiderivation we get

$$\begin{aligned} 0=\bar{V}\rfloor (\bar{\partial }\dot{\phi _s}\wedge (i\partial \bar{\partial }\phi )^n)=\overline{V(\dot{\phi _s})} (i\partial \bar{\partial }\phi )^n+n\bar{\partial }\dot{\phi _s}\wedge i\partial h^\phi \wedge (i\partial \bar{\partial }\phi )^{n-1}. \end{aligned}$$

Inserting this above we see that

$$\begin{aligned} II=- n\int i\partial \dot{\phi _t}\wedge \bar{\partial }\dot{\phi _s}\wedge (i\partial \bar{\partial }\phi )^{n-1} e^{ah^\phi }-a\int \dot{\phi _t}\overline{V(\dot{\phi _s})}(i\partial \bar{\partial }\phi )^ne^{ah^\phi }. \end{aligned}$$

Hence

$$\begin{aligned} \frac{d}{ds}\int \dot{\phi _t}(i\partial \bar{\partial }\phi )^n e^{a h^{\phi }}&= \int \ddot{\phi }_{t,s} (i\partial \bar{\partial }\phi )^n e^{a h^{\phi }}\\&\quad -\, n\int i\partial \dot{\phi _t}\wedge \bar{\partial }\dot{\phi _s}\wedge (i\partial \bar{\partial }\phi )^{n-1} e^{ah^\phi }\\&\quad +\, 2ia\int \dot{\phi _t}\mathrm{Im\, }V(\dot{\phi }_s)\wedge (i\partial \bar{\partial }\phi )^{n} e^{a h^{\phi }}. \end{aligned}$$

But the left hand side of this equality is real so the last term must be zero (which is also clear since $\phi$ is invariant under the flow of $\mathrm{Im\, }V$). We are then left with the formula in Proposition 10.1.