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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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