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Chow motif and higher Chow theory of \(G/P\). (English) Zbl 0735.14001
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 \((Y,p_ w)\) and \(L^{\otimes\ell(w)}\). Specializing to \(G=GL_{n+1}\), which has as Weyl group the symmetric group \({\mathfrak S}_{n+1}\), and \(P=P_ d\) the stabilizer of a \(d\)-plane in \(k^{n+1}\), one has a canonical isomorphism \(GL_{n+1}/P_ d@>\sim>>G_ d\), where \(G_ d\) is the Graßmannian of \(d\)-planes in \(k^{n+1}\). Setting \(W^ d=\{(\lambda_ 1,\ldots,\lambda_ d)\in\mathbb{N}^ d\mid n+1-d\geq\lambda_ 1\geq\ldots\geq\lambda_ d\geq0\}\) and \(|\lambda|=\sum^ d_{i=1}\lambda_ i\), \(\lambda\in W^ d\), one obtains a bijection between \(W^ d\) and the set \(W^{\theta_ d}\subseteq{\mathfrak S}_{n+1}\), where \(\theta_ d\) is the generating set S of transpositions of \({\mathfrak S}_{n+1}\) minus the transposition \(<d,d+1>\) (remark that one has a length function on \({\mathfrak S}_{n+1}\) relative to \(S\) and \(W^{\theta_ d}\) is defined as \(W^ \theta\) above). For the motif \(h(G_ d)\) of \(G_ d\) one obtains \(h(G_ d)=\oplus_{\lambda\in W^ d}L^{\otimes|\lambda|}\). Again one constructs a complete set of pairwise orthogonal projectors of \(\text{End}(G_ d({\mathcal E}))=CH^*(G_ d({\mathcal E})\times G_ d({\mathcal E}))\) to prove: the Chow motif \(h(G_ d({\mathcal E}))=\oplus_{\lambda\in W^ d}h(X)\otimes L^{\otimes|\lambda|}\). These results can be applied to yield calculations of (Bloch’s) higher Chow groups.

14A20 Generalizations (algebraic spaces, stacks)
14C05 Parametrization (Chow and Hilbert schemes)
14M17 Homogeneous spaces and generalizations
Full Text: DOI EuDML
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