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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 Poincaré 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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