Bounds for the anticanonical bundle of a homogeneous projective rational manifold
type: Item
instance of: scholarly article
title: Bounds for the anticanonical bundle of a homogeneous projective rational manifold
zbMATH Open document ID: 1061.14053
author: Dennis M. Snow
published in: Documenta Mathematica
publication date: 2004-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/51257
full work available at URL: 14.html
review text: 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 \({\mathbb P}_1^n\)).
reviewed by: Jörg Winkelmann
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 2095727
zbMATH Keywords: homogeneous rational manifold
zbMATH Keywords: Fano manifold
zbMATH Keywords: anticanonical bundle
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2095727