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Motivic decomposition of abelian schemes and the Fourier transform. (English) Zbl 0745.14003
For a field $$k$$, let $$S$$ be a smooth quasi-projective (connected) scheme over $$k$$. Write $${\mathcal V}(S)$$ for the category of smooth $$S$$-schemes $$\lambda :X\to S$$. The usual construction of the category of (Chow) motives over a field extends to the situation of $$S$$-schemes. One obtains categories $${\mathcal M}_ +^ 0(S)$$ and $${\mathcal M}(S)$$ of effective relative Chow motives. These are constructed as the Karoubian (pseudo-abelian) envelopes of $${\mathcal V}(S)$$ with morphisms given by the graded (resp. ungraded) correspondences $$CH^{dim(X/S)}(X\times_ SY,\mathbb{Q})=CH^{dim(X/S)}(X\times_ SY)\otimes \mathbb{Q}$$ (resp. $$CH(X\times_ SY,\mathbb{Q})=CH(X\times_ SY)\otimes \mathbb{Q})$$. For $$\lambda:X\to S$$ in $${\mathcal V}(S)$$ one writes $$R(X/S)$$ for $$(X,\hbox{id})$$ in $${\mathcal M}(S)$$, and if $$\pi_ i$$ is a projector, one writes $$R^ i(X/S)$$ for the motive $$(X,\pi_ i)$$. In particular, one has a Lefschetz motive $$L_ S=R^ 2(\mathbb{P}^ 1_ S/S)$$ such that for any $$X$$ with connected fibres of dimension $$d$$ and canonical projector $$\pi_{2d}=X\times_ Se(S)$$, $$e:S\to X$$ a section, there is an isomorphism $$R^{2d}(X/S)\rightsquigarrow L^{\otimes d}_ S$$. Localization of $${\mathcal M}^ 0_ +(S)$$ with respect to $$M\mapsto M\otimes L_ S$$ gives the category $${\mathcal M}^ 0(S)$$ of Chow motives with respect to graded correspondences. Tensoring with $$L^{\otimes -m}_ S=(L^{-1}_ S)^{\otimes m}$$ defines the twists $$M(m)$$, $$m\in \mathbb{Z}$$. Bloch’s Chow groups $$CH^ \bullet(-,j)$$ and motivic cohomology $$H^ \bullet_{\mathcal M}(-,\mathbb{Q}(j))=CH^ j(-,2j- \bullet)\otimes \mathbb{Q}$$ factor over $${\mathcal M}^ 0(S)$$ and ones has e.g. $$CH^ i(M(m),j)=CH^{i+m}(M,j)$$. $${\mathcal M}^ 0(S)$$ and $${\mathcal M}(S)$$ behave well under base change, and the functor $$R_ \ell:{\mathcal V}(S)\to D^ b(S,\mathbb{Q}_ \ell)$$, $$(\lambda:X\to S)\mapsto R\lambda_ *\mathbb{Q}_ \ell$$, extends to a $$\mathbb{Q}$$-linear functor $$R_ \ell:{\mathcal M}^ 0(S)\to D^ b(S,\mathbb{Q}_ \ell)$$ which commutes with twists and tensor products and is compatible with base extension. Here $$D^ b(S,\mathbb{Q}_ \ell)$$ denotes the bounded derived category of $$\mathbb{Q}_ \ell$$-sheaves on $$S$$.
A general problem in the theory of motives is to find a decompositon of the form $$R(X/S)=\oplus_ iR^ i(X/S)$$ for suitable projectors $$\pi_ i$$ and to describe the components $$R_ i(X/S)$$, e.g. what are their realizations? In the underlying paper a canonical functorial decomposition of $$R(A/S)$$, where $$A$$ is an abelian scheme over $$S$$, is established by means of the Fourier transform
$$F:{\mathcal M}(S)\to {\mathcal M}(S)$$, $$F=F_ A:R(A/S)\mapsto R(\hat A/S)$$, where $$\hat A/S$$ is the dual abelian scheme. $$F$$ is defined as the correspondence $$F=F_ A\in CH(A\times_ S \hat A,\mathbb{Q})$$ given by the formula $F=F_ A=ch(L)= \hbox{exp} (c_ 1({\mathcal L}))=1+{c_ 1({\mathcal L}) \over 1!}+{c_ 1({\mathcal L})^ 2 \over 2!}+\cdots,$ where $$\mathcal L$$ is a PoincarĂ© line bundle on $$A\times_ S\hat A$$ with class $$L\in Pic(A\times_ S\hat A)$$ rigidified along the zero sections, and where $$c_ 1({\mathcal L})\in CH^ 1(A\times_ S\hat A)$$ is the associated (Chern) divisor class. For the Fourier transform $$\hat F:\hat A\to \hat{\hat A}=A$$ one has $$\hat F={^ tF}$$, and with the map $$\sigma:A\to A, a\mapsto -a$$, one has $$\hat F\circ F=(-1)^ g[\Gamma_ \sigma]$$, where $$g$$ is the fibre dimension of $$A/S$$ and $$[\Gamma_ \sigma]$$ is the class of the graph of $$\sigma$$. Thus $$F$$ is an automorphism of $${\mathcal M}(S)$$ with inverse $$F^{-1}=(-1)^ g[\Gamma_ \sigma]\circ \hat F$$. Also, for an isogeny $$f:A\to B$$ between abelian schemes $$A/S$$ and $$B/S$$, one has commutativity in $${\mathcal M}(S)$$:
(i) $$F_ A\circ f^*=\hat f_ *\circ F_ B$$ and
(ii) $$F_ B\circ f_ *=\hat f^*\circ F_ A$$.
$$F$$ defines a homomorphism $$F_{CH}:CH(A,\mathbb{Q})\to CH(\hat A,\mathbb{Q})$$ by $$F_{CH}(\xi)=p_{2*}(p^*_ 1(\xi)\cdot F)$$ and (i) and (ii) carry over to corresponding properties of $$F_{CH}$$. The $$F_{CH}$$ play a main role in the proof of the final result of the paper:
Theorem 1. Let $$\lambda :A\to S$$ be an abelian scheme of fibre dimension $$g$$ and let $$n:A\to A$$ be multiplication by $$n$$. Then the diagonal $$\Delta=\Delta(A/S)$$ has a unique decomposition $$\Delta=\sum^{2g}_{i=0}\pi_ i$$ in $$CH^ g(A\times_ SA,\mathbb{Q}),$$ where the $$\pi_ i$$ are pairwise orthogonal idempotents, such that $$(id_ A\times n)^*\pi_ i=n^ i\pi_ i$$ for all $$n\in \mathbb{Z}$$. Moreover, $$[^ t\Gamma_ n]\circ \pi_ i=\pi_ i\circ [^ t\Gamma_ n]=n^ i\pi_ i$$.
Corollary 2. (i) The motive $$R(A/S)$$ decomposes as $$(*)$$ $$R(A/S)=\bigoplus^{2g}_{i=0}R^ i(A/S)$$, where $$R^ i(A/S)=(R(A/S),\pi_ i)$$ are the relative Chow motives in $${\mathcal M}^ 0(S)$$ determined bz $$\pi_ i$$. $$n^*$$ acts by multiplication with $$n_ i$$ on $$R^ i(A/S)$$.
(ii) For the $$\ell$$-adic realization of $$R(A/S)$$ one obtains $$R^ i_ \ell(R^ j(A/S))=0$$, for $$i\neq j$$, $$R^ i_ \ell(R^ j(A/S))=R^ j\lambda^*\mathbb{Q}_ \ell$$, for $$i=j$$.
(iii) The decomposition $$(*)$$ induces a canonical splitting in $$D^ b(S,\mathbb{Q}_ \ell):$$ $R\lambda_ *\mathbb{Q}_ \ell\cong\bigoplus^{2g}_{i=0}R^ i\lambda_ *\mathbb{Q}_ \ell [-i].$ The paper closes with a remark on the motive $$H_ \chi$$ attached to a Hecke character $$\chi$$ of a number field $$K$$ with values in a number field $$E$$. In this case, there exists a Chow motive $$M'_ \chi$$ over $$S=\hbox{Spec}(K)$$ with coefficients in an extension $$E'$$ of $$E$$ such that for the absolute Hodge realization $$M'{}^{a.H.}_ \chi$$ of $$M'_ \chi$$ one has $$M'{}^{a.H.}_ \chi=H_ \chi\times_ EE'$$.

##### MSC:
 14A20 Generalizations (algebraic spaces, stacks) 14K05 Algebraic theory of abelian varieties 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14K02 Isogeny 14C05 Parametrization (Chow and Hilbert schemes)
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