Chow motif and higher Chow theory of \(G/P\)
type: Item
instance of: scholarly article
title: Chow motif and higher Chow theory of \(G/P\)
zbMATH Open document ID: 0735.14001
DOI: 10.1007/BF02568384
author: Bernhard Köck
published in: Manuscripta Mathematica
publication date: 1992-06-25T00:00:00Z
full work available at URL: https://eudml.org/doc/155595
review text: For a field \(k\) let \(\mathbb{P}^ n\) denote the projective \(n\)-space over \(k\). Then it is well known that the Chow motif of \(\mathbb{P}^ n, h(\mathbb{P}^ n)\), is given by \(h(\mathbb{P}^ n)=1\oplus L\oplus L^{\otimes 2}\oplus\cdots\oplus L^{\otimes n}\), where \(L=(\mathbb{P}^ 1, [1\times e])\) is the Tate (or Lefschetz) motif. More generally, using Manin's identity principle, one shows that, for a vector bundle \(\mathcal E\) of \(\hbox{rank }n+1\) on the smooth projective \(k\)-variety \(X\) the motif of the projective bundle \(\mathbb{P}({\mathcal E})\) is given by \(h(\mathbb{P}({\mathcal E}))=\oplus^ n_{i=0} h(X)\otimes L^{\otimes i}\), where \(h(X)\) is the motif of \(X\). In this paper these results are generalized to homogeneous spaces \(Y=G/P\), \(G\) a \(k\)-split reductive linear algebraic group defined over \(k\) and \(P\) a parabolic subgroup of \(G\), and to the Graßmann bundle \(G_ d({\mathcal E})\) of \(d\)-planes in the \(\hbox{rank }n+1\) vector bundle \(\mathcal E\) on a smooth projective \(k\)-variety \(X\). For such \(G\) with maximal \(k\)-split torus \(T\) and Borel \(k\)-subgroup \(B\) containing \(T\), let \(S\) be the set of reflections in the Weyl group corresponding to the set of simple \(B\)-positive roots, and let \(\ell:W\to\mathbb{N}\) be the length function (relative to \(S\)). For a fixed subset \(\theta\) of \(S\), let \(W_ \theta\) denote the subgroup of \(W\) generated by \(\theta\) and let \(W^ \theta=\{w\in W\mid\ell(ws)=\ell(w)+1\), for all \(s\in\theta\}\). For the corresponding parabolic subgroup \(P=P_ \theta=BW_ \theta B\) let \(Y\) be the smooth projective \(k\)-variety \(G/P\). Then the Chow motif of \(Y\) is given by \(h(Y)=\oplus_{w\in W^ \theta} L^{\otimes\ell(w)}\). The proof is based on an explicit construction of \(\text{End}(h(Y))\) via the Bruhat decomposition of \(Y\) and of a complete set of pairwise orthogonal projectors \(\{p_ w\}_{w\in W^ \theta}\) in \(\text{End}(h(Y))\), and an isomorphism between…
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 8987
zbMATH Keywords: Chow motif
zbMATH Keywords: Tate motif
zbMATH Keywords: Lefschetz motif
zbMATH Keywords: homogeneous space
zbMATH Keywords: Bruhat decomposition
zbMATH Keywords: Grassmannian of d-planes
zbMATH Keywords: higher Chow groups
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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OpenAlex ID: W2057315799
description: scientific article; zbMATH DE number 8987