Quantized volume comparison for Fano manifolds
Kewei Zhang
TL;DR
The paper quantizes Fujita's volume comparison for Fano manifolds by translating volume bounds into discrete jet-space bounds via the dimensions $\dim H^0(X,-mK_X)$. It proves that for K-semistable (and certain) Fanos, these dimensions are bounded above by the corresponding dimensions for $\mathbb{P}^n$, with equality characterizing $X\cong \mathbb{P}^n$, and strengthens the result through a localized $\delta_m$-invariant criterion. An elementary linear-algebraic argument underpins the main results, yielding new insights into linear systems and jet-separation, and the paper provides partial results in low dimensions (notably $n\le3$ and a conditional $n=4$ case). These findings connect stability notions to discrete volume-type invariants and open paths for further study of quantized invariants like $\delta_m(-K_X)$ in higher dimensions.
Abstract
A result of Kento Fujita says that the volume of a Kähler-Einstein Fano manifold is bounded from above by the volume of the projective space. In this short note we establish quantized versions of Fujita's result.