A vanishing theorem ``à la Kawamata-Viehweg''
type: Item
instance of: scholarly article
title: A vanishing theorem ``à la Kawamata-Viehweg''
zbMATH Open document ID: 0820.14012
DOI: 10.1007/BF02567622
author: Laurent Manivel
published in: Manuscripta Mathematica
publication date: 1994-10-05T00:00:00Z
full work available at URL: https://eudml.org/doc/155969
review text: In this article, the author considers nef vector bundles (i.e., the ones whose tautological bundle is nef) of arbitrary rank over smooth projective varieties. Let \(E\) be an algebraic vector bundle of rank \(r\) on a smooth projective variety \(X\) of dimension \(n\). Let \(s = \{i | 0 \leq i \leq r\}\), Fl\(_ s(E)\) = flag variety on \(E\) of form \(s\), \(N_ s\) = relative dimension on \(X\), and \(\rho =\) a partition of length \(l(\rho)\) \((\leq r)\). Now Fl\(_ s(E)\) admits a line bundle \({\mathfrak L}^ \rho\) whose direct image on \(X\) is the fibre \(S^ \rho (E)\) associated to \(E\) for the irreducible representation of \(GL_ r (\mathbb{C})\) defined by \(\rho\). Using the following result: \[ {c_ 1 ({\mathfrak L}^ \rho)^{n + N_ s} \over (n + N_ s)!} = C_{\rho,r} \sum_{| \pi | = n,l (\pi)\leq l(\rho)} \int_ X {s_ \pi (\rho) s_ \pi (E) \over \prod^{l(\rho)}_{k = 1} (r + \pi_ k - k)!} \] where \(C_{\rho,r} \in \mathbb{Z}^ +\) and \(s_ \pi\) is the Schur polynomial associated to the partition \(\pi\) and \(s_ \pi (E) = \text{det} (c_{\pi^*_ i - i + j} (E))\), \(\pi^*\) being the conjugate partition of \(\pi\) in the Young diagram, the author notes that the coefficient of \(s_ \pi (E)\) is positive, and that \({\mathfrak L}^ \rho\) is nef and big if and only if the Schur polynomial admits strictly positive values on \(X\), whence the Kawamata-Viehweg vanishing theorem gives the following main result of the paper: If \(E\) is an algebraic vector bundle of rank \(r\), and \(E\) is nef on a smooth projective variety \(X\) of dimension \(n\), and if \({\mathfrak L}\) is a nef line bundle on \(X\) with a partition \(\mu\) such that \(\int s_ \mu (E) > 0\) (which is weaker than \({\mathfrak L}\) big on \(X)\), then for all partitions \(\rho\) of length \(l(\rho) \geq l(\mu)\), \[ H^ q (X,K_ X \otimes S^ \rho E \otimes (\text{det} E)^ l \otimes {\mathfrak L}) = 0,\quad\text{if}\quad q > 0\quad\text{and}\quad l \geq l (\rho). \] Along the way, the author discusses cl…
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 624944
zbMATH Keywords: nef vector bundles
zbMATH Keywords: Kawamata-Viehweg vanishing theorem
zbMATH Keywords: self- intersections of line bundles on flag varieties
zbMATH Keywords: Euler-Poincaré characteristics
reviewed by: Harry D'Souza
MaRDI profile type: Publication
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OpenAlex ID: W2317667603
description: scientific article; zbMATH DE number 624944