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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.

MSC:
14A20 Generalizations (algebraic spaces, stacks)
14C05 Parametrization (Chow and Hilbert schemes)
14M17 Homogeneous spaces and generalizations
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