The paper under review studies the upper bound of volumes of Fano manifolds. In dimension less than 4, it is known by classification results that the projective space is the (only) Fano manifold with the largest volume in each dimension. This is no longer true for in dimension at least 4. However, R. J. Berman and B. Berndtsson [“The projective space has maximal volume among all toric Kähler-Einstein manifolds”, arXiv:1112.4445] conjectured that this is the case if we only concentrate on Kähler-Einstein Fano manifolds, that is, Fano manifolds admitting Kähler-Einstein metrics, which is an important class of Fano manifolds in the study of differential geometry of Fano manifolds. Berman and Berndtsson confirmed this conjecture for toric ones.
As the main result of this paper, it is shown that for any Kähler-Einstein Fano manifold \(X\) of dimension \(d\), the volume inequality \((-K_X)^d\leq (d+1)^d\) holds. Moreover, equality holds if and only if \(X\) is the projective space.
Although the existence of Kähler-Einstein metrics is a differential geometric property, the proof of this paper is purely algebraic. Instead of considering the existence of Kähler-Einstein metrics, one can consider Ding semi-stability, which is a necessary condition, but more algebraic. It is shown that for any Ding semistable (possibly singular) \(X\) of dimension \(d\), the volume \((-K_X)^d\leq (d+1)^d\). Moreover assuming \(X\) to be smooth, the equality holds if and only if \(X\) is the projective space.
The idea of the proof is to interpret the original definition of Ding semistability (with uses test configuration and Ding invariants) in terms of so-called \(\beta\)-invariants, which involve log canonical thresholds and integration of volume functions. This intepretion here is an incomplete version (but sufficient for the use of this paper) which is completed by the same author in a later paper, which turns out to be very useful in the study of K-semistability.