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For an unconditional theory of motives. (Pour une théorie inconditionnelle des motifs.) (French) Zbl 0874.14010
It has been known, ever since its invention by Grothendieck in the late 1960’s, that the theory of motives hinges on two problems:
(1) Foundations. This boils down essentially two one of Grothendieck’s standard conjectures on algebraic cycles which says that numerical and homological equivalence should coincide. At the very heart of this problem is
(2) Construction of enough motivated (i.e. algebraic) cycles. Here one is confronted with the Hodge and Tate conjectures, as well as the conjecture on the algebraicity of the Lefschetz involution \(*_L\).
In the present paper an unconditional theory of motives is proposed, essentially by the formal adjunction of \(*_L\) to the algebra of algebraic correspondences to obtain the algebra of ‘motivated cycles’. More precisely, let \(K\) be a field and denote by \(\mathcal V\) a full subcategory of the category of smooth projective \(K\)-schemes, stable under products, disjoint sums and connected components. The objects of \(\mathcal V\) are called base pieces. One fixes a Weil cohomology \(H^{\bullet}\) on \(\mathcal V\) having coefficients in a field \(F\) of characteristic zero and verifying the ‘hard Lefschetz theorem’ for the objects \(X\) of \(\mathcal V\), i.e. for a smooth projective \(K\)-scheme \(X\) of pure dimension \(d\) and a hyperplane section with class \(L\in H^2(X)\) the \((d-i)\)-fold cup product with \(L\) induces an isomorphism \(L^{d-i}:H^i(X)\to H^{2d-i}\) for any \(i\leq d\). The corresponding Lefschetz involution is denoted by \(*_L\). Closely related one has the Hodge involution \(*_H\). For a subfield \(E\) of \(F\) one defines a motivated cycle with coefficients in \(E\) as an element in \(H^{\bullet}\) of the form \(\text{pr}^{XY}_{X *}(\alpha\cup *\beta)\), where \(\text{pr}^{XY}_X:X\times Y\rightarrow X\) is the projection, \(*\) is either \(*_L\) or \(*_H\), \(\alpha\) and \(\beta\) are algebraic cycles on \(X\times Y\) with coefficients in \(E\) for arbitrary \(Y\) in \(\mathcal V\), and the subscript \(*=*_{XY}\) is with respect to the class \(\eta_{X\times Y}=[X]\otimes\eta_Y+\eta_X\otimes[Y]\) with any hyperplane classes \(\eta_X\) and \(\eta_Y\) on \(X\) and \(Y\), respectively. The collection \(A_{\text{mot}}(X)_E\) of motivated cycles on \(X\) with coefficients in \(E\) is shown to be an \(E\)-subalgebra of \(H^{\bullet}\) (with respect to the cup product), graded by ‘codimension’ such that for any classical cohomology \(H\) there is an injective linear map \(\text{cl}_H:A^{\bullet}_{\text{mot}}(X)_E\to H^{2\bullet}(X)\) which extends the cycle class map for algebraic cycles (modulo homological equivalence), and which admits the (\(f^*\), \(f_*\))-formalism for morphisms \(f\), related by the projection formula. In a straightforward way one constructs via the \(A^{\bullet}_{\text{mot}}(X)_E\)’s graded spaces \(C^{\bullet}_{\text{mot}}(X\times Y)_E\) of motivated \(E\)-correspondences from \(X\) to \(Y\). Motivated correspondences can be composed such that degrees are added. In particular, \(C^{\bullet}_{\text{mot}}(X,X)_E\) is a graded \(E\)-algebra. It contains the involutions \(*_L\) and \(*_H\), and also the Künneth projectors \(\pi^i_X\). Furthermore, if the characteristic of \(K\) is zero, one shows that the collection of the de Rham and the \(\ell\)-adic cohomology classes of any motivated cycle is an absolute Hodge cycle in the sense of Deligne. Two motivated cycles \(x,y\in A_{\text{mot}}(X)_E\) are said to be numerically equivalent, written \(x\equiv y\), if for any \(z\in A_{\text{mot}}(X)_E\) one has \(\displaystyle{\int_X(x-y)\cup z=0}\). It can be shown that \(\equiv\) is an adequate equivalence relation. Thus, one may define the quotients \(\overline A_{\text{mot}}^{\bullet}(X)_E:=A_{\text{mot}}^{\bullet}(X)_E/\equiv\) and \(\overline C_{\text{mot}}^{\bullet}(X)_E:=C_{\text{mot}}^{\bullet}(X)_E/\equiv\).
Assume \(K\) has characteristic zero and \(H^{\bullet}\) is any classical cohomology. Using motivated correspondences one may define the category of motives modeled on \({\mathcal V}\), written \({\mathcal M}(\mathcal V)\), in the usual way: Objects are triples \(M=(X,n,q)\) (also written \(M=qh(X)(n)\)) with \(X\) in \(\mathcal V\), \(n\) a continuous integer valued function on \(X\) (giving rise to the Tate twist), and \(q\) a projector, i.e. a degree zero idempotent correspondence from \(X\) to itself; morphisms \(\text{Hom}((X,n,q),(Y,m,p))=pC^{m-n}_{\text{mot}}(X,Y)q\). For \(X\) in \(\mathcal V\) one defines its motivic cohomology \(h(X):=(X,0,\text{id})\). This gives a contravariant functor \(h:{\mathcal V}\rightarrow{\mathcal M}(\mathcal V)\), where to a morphism one associates the homological equivalence class of the transpose of its graph.
The main result for \(\mathcal M(\mathcal V)\) is that it is a graded, semi-simple, polarized tannakian category over \(\mathbb{Q}\), i.e. it is \(\otimes\)-equivalent to the category of representations of an algebraic gerb (with the additional properties of weights and a positive symmetric bilinear Hodge form). Taking for \(\mathcal V\) the category of smooth projective \(K\)-schemes one obtains the tannakian category \({\mathcal M}_K\) of motives over \(K\). It is possible to develop unconditionally a motivic Galois theory for \({\mathcal M}({\mathcal V})\).
The motivic Galois group \(\mathbf{G}_{{\mathcal M}({\mathcal V})}\) of \({\mathcal M}({\mathcal V})\) is defined as the group of tensor automorphisms of the Betti realization functor on \({\mathcal M}({\mathcal V})\). One has an exact sequence \[ 1\rightarrow\mathbf{G}_{\overline{K}}\rightarrow\text\textbf{G}_K\rightarrow\text{Gal} (\overline{K}/K)\rightarrow 1, \] and for any prime \(\ell\) there is a canonical continuous splitting \(\rho_{\ell}:\text{Gal}(\overline{K}/K)\rightarrow\mathbf{G}_K(\mathbb{Q}_{\ell})\). With respect to the construction of motivated cycles, Hironaka’s desingularization theorem, Deligne’s theorem of the fixed part and the properties of \(\mathcal M(\mathcal V)\) imply a motivated variant of a result conjectured by Grothendieck on the invariance of the notion of algebraic cycle under flat deformation:
Let \(S\) be a reduced connected scheme of finite type over \(\mathbb{C}\) and let \(f:X\rightarrow S\) be a smooth projective morphism. Let \(\xi\) be a section of \(\text{R}^{2p}f_*\mathbb{Q}(p)\) on \(S^{\text{an}}\). Then, if for a point \(s\in S(\mathbb{C})\) the fibre \(\xi_s\) is motivated, \(\xi_t\) is motivated for any \(t\in S(\mathbb{C})\). Furthermore, if \(S\) is smooth projective, \(\xi\) comes from a motivated cycle \(\xi_X\) on \(X\).
Four corollaries of this result are discussed. One of these says that for any field \(K\) of characteristic zero, every absolute Hodge cycle on an abelian \(K\)-variety is motivated. This sheds some new light on the Hodge conjecture for abelian varieties. Another one says that the motif of a projective \(K3\) surface or a cubic hypersurface in \(\mathbb{P}^n\), \(n\leq 6\), is isomorphic to a motif coming from an abelian variety.
A motif with integer coefficients is defined as the data of a motif \(M\) in \({\mathcal M}({\mathcal V})\) and a \({\mathbb Z}\)-lattice \(\Lambda\) in \({\mathcal H}_B(M)\), where \({\mathcal H}_B\) denotes the Betti realization functor on \({\mathcal M}({\mathcal V})\), such that \(\Lambda\otimes{\mathbb Z}_{\ell}\) is stable under \(\text{Gal}(\overline{K}/K)\) for every prime \(\ell\). These motives are the objects of a rigid \({\mathbb Z}\)-linear tensor category \({\mathcal M}'({\mathcal V})[{\mathbb Z}]\). Demanding that \(\Lambda\) be an abelian group of finite type with Galois action on \(\Lambda\otimes{\mathbb Z}_{\ell}\) such that \(\Lambda\otimes{\mathbb Q}\cong{\mathcal H}_B(M)\) and such that the ensueing isomorphism \(\Lambda\otimes{\mathbb Q}_{\ell}\cong{\mathcal H}_B(M)\otimes{\mathbb Q}_{\ell}\) is Galois invariant, this defines another \({\mathbb Z}\)-linear tensor category \({\mathcal M}({\mathcal V})[{\mathbb Z}]\) whose objects are quotients of those in \({\mathcal M}'({\mathcal V})[{\mathbb Z}]\). Both these categories are \(\otimes\)-equivalent to the category of representations (of finite type over \({\mathbb Z}\)) of an affine group scheme which is flat over \({\mathbb Z}\), the first one being the subcategory of the second one consisting of torsion free objects.
The last section is concerned with the construction and properties of motives in characteristic \(p\) and specialization. Again, fix a Weil cohomology \(H^{\bullet}\) on \(\mathcal V\). The construction is analogous to the one in characteristic zero, except that one takes for the morphisms the spaces \(\overline{p}\overline{C}^{m-n}_{\text{mot}}(X,Y)_Q\overline{q}\), with \(\overline{q}\in\overline{C}^0_{\text{mot}}(X,X)_Q\) and \(\overline{p}\in\overline{C}^0_{\text{mot}}(Y,Y)_Q\). Here \(Q\) is the subfield of \(F\) generated by the values \(\displaystyle{\int_X\alpha\cup *\alpha}\), where \(\alpha\) runs over the algebraic cycles on \(X\). This construction leads to a tannakian category over the subfield \(Q\) of \(F\). One has a fibre functor with values in an extension of \(Q\). However, there is no natural \(H\)-realization.
The paper closes with an appendix on numerical and homological equivalence. In particular, it is shown that the fact that the category of motives is abelian and the grading on \(H^{\bullet}(X)\) comes from a grading of the motif \(h(X)\) are equivalent to the fact that numerical equivalence coincides with homological equivalence. This strengthens a well-known result of U. Jannsen in one direction.

MSC:
14F99 (Co)homology theory in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E20 Relations of \(K\)-theory with cohomology theories
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