Numerically effectiveness and principal bundles on Kähler manifolds
type: Item
instance of: scholarly article
title: Numerically effectiveness and principal bundles on Kähler manifolds
zbMATH Open document ID: 1156.32009
author: Indranil Biswas
author: Georg Schumacher
published in: Annals of Global Analysis and Geometry
publication date: 2008-09-10T00:00:00Z
review text: Let \((X,\omega)\) be a compact connected Kähler manifold, \(G\) a connected simple linear algebraic group over \(\mathbb C\) and fix a proper parabolic subgroup \(P\subset G\). Moreover, consider a principal \(G\)-bundle \(E_G\) over \(X\) and any strictly antidominant character \(\chi\) of \(P\) (recall that a character \(\chi\) of \(P\) is said to be strictly antidominant if the associated line bundle over \(G/P\) is ample). Finally, a holomorphic vector bundle \(F\to X\) is said to be pseudostable, if there exists a filtration of holomorphic subbundles of \(F\) \[ 0=F_0\subset F_1\subset\cdots\subset F_{\ell-1}\subset F_\ell=F \] such that for each \(i=1,\dots,\ell\), the quotient \(F_i/F_{i-1}\) is a polystable vector bundle (i.e. a direct sum of stable vector bundles) with \(\mu(F_i/F_{i-1})=\mu(F)\), where \(\mu(\bullet)\) is the slope. In this paper the authors investigate the nefness of the line bundle \(E_G(\chi)\) over the quotient space \(E_G/P\), which is obtained by taking the quotient of \(E_G\times\mathbb C\) by \(P\) acting on the right as follows: the action of \(g\in P\) sends any \((z,c)\in E_G\times\mathbb C\) to \((zg,\chi(g^{-1})c)\). More precisely, they prove the equivalence of the following five facts. \(\bullet\quad\) The line bundle \(E_G(\chi)\to E_G/P\) is nef. \(\bullet\quad\) The \(G\)-bundle \(E_G\) is pseudostable and the second Chern class \(c_2(\operatorname{ad}(E_G))\in H^4(X,\mathbb Z)\) vanishes, where \(\operatorname{ad}(E_G)\) is the holomorphic vector bundle over \(X\) associated to the principal \(G\)-bundle \(E_G\) for the adjoint action of \(G\) on its own Lie algebra. \(\bullet\quad\) The vector bundle \(\operatorname{ad}(E_G)\) is numerically flat. \(\bullet\quad\) Take any triple \((Y,\omega_Y,\beta)\), where \((Y,\omega_Y)\) is a reduced irreducible compact Kähler analytic space of dimension \(m\) and \(\beta\colon Y\to X\) a holomorphic map, and let \(E_P\subset\beta^*E_G\) be any holomorphic reduction of the structu…
reviewed by: Simone Diverio
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 5324299
zbMATH Keywords: numerically effectiveness
zbMATH Keywords: Kähler manifold
zbMATH Keywords: principal bundle
MaRDI profile type: Publication
full work available at URL: s10455-007-9099-z
OpenAlex ID: W2081918214
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DOI: 10.1007/S10455-007-9099-Z
description: scientific article; zbMATH DE number 5324299