Kaixuan Lyu

Quantized Volume Comparison for Fano Manifolds, II

In this note, the second author's quantized volume comparison conjecture is solved: If $X$ is a $K$-semistable Fano manifold of dimension $n$, then for every integer $m\geq1$, \[ h^0(X,-mK_X)\leq h^0(\mathbb P^n,-mK_{\mathbb P^n}) =\binom{n+m(n+1)}{n}, \] and equality for one $m$ characterizes projective space. Somewha...