Bounds for the Anticanonical Bundle of a Homogeneous Projective Rational Manifold

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: \newcommand{\binom}[2]{{{#1}\choose{#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.

2000 Mathematics Subject Classification: Primary 14M17; Secondary 14M15, 32M10

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