Optimal bounds for the volumes of K\"ahler-Einstein Fano manifolds

TL;DR

This paper establishes an upper bound for the volume of Fano manifolds with K"ahler-Einstein metrics, showing the maximum is achieved only by projective space, thus characterizing the extremal case.

Contribution

It proves a sharp volume bound for K"ahler-Einstein Fano manifolds and characterizes the case of equality as projective space.

Findings

Anti-canonical volume of Fano manifolds is at most (n+1)^n.

Equality in volume bound characterizes projective space.

Provides a geometric criterion for extremal Fano manifolds.

Abstract

We show that any $n$-dimensional Fano manifold $X$ admitting K\"ahler-Einstein metrics satisfies that the anti-canonical volume is less than or equal to the value $(n+1)^n$. Moreover, the equality holds if and only if $X$ is isomorphic to the $n$-dimensional projective space.