## Bounds for the anticanonical bundle of a homogeneous projective rational manifold

### Summary

Summary: The following bounds for the anticanonical bundle $K_X^{*} = \det T_X$ of a complex homogeneous projective rational manifold $X$ of dimension $n$ are established: newcommandbinom[2]#1choose#2 $$3^n \le \dim H^0(X,K_X^{*}) \le \binom{2n+1}n\quad\mathrm{and}\quad 2^n n! \le \deg K_X^{*} \le (n+1)^n$$ 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.

### Mathematics Subject Classification

14M17, 14M15, 32M10