Bounds for the anticanonical bundle of a homogeneous projective rational manifold

Abstract

The following bounds for the anticanonical bundle $K_X^{\ast }=\det T_X$ of a complex homogeneous projective rational manifold $X$ of dimension $n$ are established: newcommandbinom[2]#1choose#2 $3^n\le \dim H^0(X,K_X^{\ast })\le \left(\genfrac{}{}{0ex}{}{2n+1}{n}\right)\mathrm{and}{2}^{n}n!\le \mathrm{deg}{K}_X^{\ast }\le (n+1{)}^n$ with equality in the lower bounds if and only if $X$ is a flag manifold and equality in the upper bounds if and only if $X$ is complex projective space. None of these bounds holds for general Fano manifolds.