Bounds for the anticanonical bundle of a homogeneous projective rational manifold
Let \(X\) be a homogeneous rational projective manifold, i.e., a complex manifold which is isomorphic to a quotient \(G/P\) of a semisimple complex Lie group \(G\) by a parabolic subgroup \(P\). Let \(K_X^*\) be the anticanonical bundle. This is the line bundle whose sections are holomorphic \(n\)-forms where \(n=\dim(X)\). There are two important numbers associated to this bundle: the dimension of the vector space \(H^0(X,K_X^*)\) of sections and the ``degree'' \(\deg(K_X^*)\) which can be defined via \(\deg(K_X^*)=c_1(K^*)^n\in H^{2n}(X,{\mathbb Z})\simeq {\mathbb Z}\). For any given homogeneous rational manifold \(G/P\) this numbers can be calculated in terms of combinatorial data related to the root system of \(G\), however, this relationship is technically intricate. The purpose of this article is to deduce precise lower and upper bounds for these numbers. It turns out that \(3^n\leq\dim H^0(X,K^*_X)\leq \begin{pmatrix} 2n+1\cr n \cr \end{pmatrix}\) and \(2^nn!\leq \deg(K^*_X) \leq (n+1)^n\) whenever \(X\) is a homogeneous rational manifold. The upper bounds are attained for \(X\simeq{\mathbb P}^n\) while the lower bounds hold for any flag manifold, i.e. for any quotient of a semisimple Lie group \(G\) by a Borel subgroup \(B\) (for instance \({\mathbb P}_1^n\)).
scientific article; zbMATH DE number 2095727