The theory of motives is intimately related to the existence of algebraic cycles.
E.g. Grothendieck’s standard conjectures demand that the Künneth components of the cohomology class of the diagonal \(\Delta\subset X\times X\), \(X\) a smooth projective variety over a field \(k\), be algebraic. In general it has been impossible to construct (enough) algebraic cycles on algebraic varieties and one is left with deep conjectures, e.g. over the complex numbers \(\mathbb{C}\) one has the still open Hodge conjecture. To get a better understanding of cycles and cycle maps the author was led to introduce the now called Deligne cohomology, i.e. an extension of the Hodge group (of algebraic cohomology classes as predicted by the Hodge conjecture) by the intermediate jacobian introduced by Griffiths:
A codimension \(d\) cycle \(Z\) on a smooth projective (complex) variety \(X\) lives in a natural way in this extension \(E_ d\), but as a matter of fact one has no clear idea of the image of the cycle group (Chow group) in this \(E_ d\). Also, there is a mysterious map, the Abel-Jacobi map, from the subgroup of homologically trivial cycles to the intermediate jacobian \(J^ d\). The description of the cycle problem and Deligne cohomology can elegantly be done in terms of mixed Hodge structures: the Hodge group corresponding to \(d\)-codimensional cycles is given by \(\operatorname{Hom} (\mathbb{Z}, H^{2d} (X)(d))\) and the intermediate jacobian \(J^ d\) becomes \(\text{Ext}^ 1 (\mathbb{Z}, H^{2d-1} (X)(d))\) in the category of mixed Hodge structures, thus revealing the motivic nature of the problem of coming to terms with algebraic cycles. In some cases one can prove the Hodge conjecture using Griffiths’ intermediate jacobians and Lefschetz’ method for surfaces to lower inductively the dimension of the variety, but this does not work in general.
The most familiar and best known (accepting Grothendieck’s standard conjectures) category of motives is the one of pure motives, defined by smooth projective varieties defined over a field \(k\) and their cohomology. This category \({\mathcal M}(k)\) must be (semi-simple) tannakian, i.e. it is \(\mathbb{Q}\)-linear abelian with finite dimensional Hom’s, and it has tensor products and internal Hom’s \(\ldots\) in such a way that it is equivalent to the category of linear representations of an affine group scheme (in characteristic zero) or a suitable gerbe (in characteristic \(p>0\)) \(G= \underline {\operatorname{Aut}}^ \otimes (\omega)\) for a fibre functor \(\omega\). This motivic Galois group \(G\) is not uniquely determined because there may be more fibre functors, but two such groups are inner forms of each other. In general \(G\) will be huge. One can do algebraic geometry in tannakian categories and define e.g. group schemes in such categories. The motivic Galois group \(G\) can now be deduced from a truely motivic version \(G_{\text{mot}}\) by applying the fibre functor \(\omega\). For the category of mixed motives \({\mathcal {MM}} (k)\) (or simply written \({\mathcal M}(k)\)), one wants every object \(M\) to admit a ‘weight filtration’ \(W\) such that the graded pieces \(\text{Gr}^ W_ n (M)\) are pure motives of weight \(n\). As in the case of smooth projective varieties \(X\) giving rise to pure motives \(H^ i_{\text{mot}} (X)\), in the mixed case one wants, for any algebraic variety \(X\) over \(k\), motivic cohomology groups \(H^ i_{\text{mot}} (X)\). These must have ‘realizations’ in all the usual cohomology theories. Again one wants \({\mathcal M} (k)\) to be tannakian. In general, \({\mathcal M} (k)\) will not be semi-simple, but for \(k= \mathbb{F}_ q\) one expects that every motive is a direct sum of pure motives and the motivic Galois group \(G\) should contain a Frobenius element \(F\) such that the powers of \(F\) are dense in \(G\). In particular, \(G\) must be commutative.
An instructive example is provided by the category of abelian varieties (over \(k\)) up to isogeny, in particular, the jacobians of smooth projective curves. For such a curve one may consider its homology \(H_ 1\) as a ‘realization’ of its jacobian. This leads to particular motives of weight \(-1\). More generally, one may look at not necessarily complete smooth curves. One is led to the theory of 1-motives. These should be identified with particular mixed motives.
Generalizing the situation of abelian varieties one may try to find moduli spaces for motives. Again, it is suggestive to look first at the Hodge realization and the associated variation of (polarized) Hodge structures, and apply Griffiths’ theory. However, one is faced with a serious difficulty: Do all these Hodge structures arise from motives? If the answer is yes, one is led to believe that the Shimura varieties, the complex points of which give the moduli of Hodge structures, are moduli spaces of motives. This is discussed in some detail.
Grothendieck already saw that for any \(X\), \(H^ i (X)\) must decompose in terms of subquotients of the cohomology of smooth projective varieties, and therefore \(H^ i_{\text{mot}} (X)\) gives rise to a class in the Grothendieck group of pure motives. One may go further and look for a weight filtration \(W\) on any motive \(M\) such that \(\text{Gr}^ W_ i (M)\) is pure of weight \(i\). The first example is provided by 1-motives, in particular their associated mixed Hodge structures. If one wants to take a variation of mixed Hodge structures as a starting point for mixed motives, care must be taken whenever the transversality condition is non- trivial in Hodge theory. Grothendieck’s six operations and vanishing cycles were introduced into Hodge theory by M. Saito in his work on mixed Hodge modules. This was partially inspired by \(\ell\)-adic theory and a motivic point of view. It is expected that a good weight filtration will show up only in the context of the formalism of derived categories and perverse sheaves.
Another situation where one may expect a motivic formalism is concerned with the fundamental group. Let \(X_ 0\) be an algebraic variety defined over a field \(k\) and let \(X= X_ 0\times_ k \overline{k}\). For \(x\in X_ 0(k)\), the profinite fundamental group \(\pi_ 1 (X,x)\) is acted upon by \(\text{Gal} (\overline{k}/k)\), but this does not imply an obvious motivic interpretation of \(\pi_ 1(X,x)\). It seems easier to consider its pro-\(\ell\)-completion \(\pi_ 1 (X,x)^ \wedge_ \ell\). Anyhow, one wants a motivic group scheme \(\pi_ 1(X,x)_{\text{un}}\) over \(k\) with suitable properties, e.g. the functor ‘fibre in \(x\)’ should give an equivalence between the category of ‘flat \(X_ 0\)-motives that are iterated extensions of inverse images of motives over \(k\)’ and the category of ‘motives over \(k\) with an action by \(\pi_ 1 (X,x)_{\text{un}}\)’. These ideas can be further generalized and applied to Hodge theory.
In the usual cohomology theories \(H^ i(X)\) can be written as \(H^ i R\Gamma(X)\), where \(R\Gamma (X)\) is a functor with values in a triangulated category \({\mathcal D}\) with \(t\)-structure. \(H^ \bullet\) takes values in the heart of this \(t\)-structure. Concerning (mixed) motives one is led to conjecture the existence of a triangulated category \({\mathcal D} (k)\) with \(t\)-structure and with heart the category \({\mathcal M} (k)\) of mixed motives over \(k\). One also wants a functor \(R \Gamma_{\text{mot}}\) with values in \({\mathcal D} (k)\) such that the \(R\Gamma\) as above are the ‘realizations’ in the various cohomology theories. \({\mathcal M}(k)\) is supposed to be tannakian with fibre functor given by \(\ell\)-adic cohomology. \({\mathcal D} (k)\) should allow ‘Tate twists’. For an algebraic \(k\)-variety \(X\) one defines its absolute cohomology \(H^ i_{\text{abs}} (X):= \operatorname{Hom}_{{\mathcal D} (k)} (1,R \Gamma_{\text{mot}} (X) [i])\), where 1 is the unit motive. One has a spectral sequence \(E_ 2^{pq}= H^ p_{\text{abs}} (H^ q_{\text{mot}} (X))\to H^{p+q}_{\text{abs}} (X)\), where \(H^ p_{\text{abs}} (M)= \text{Ext}^ p (1,M)= \operatorname{Hom}_{{\mathcal D}(k)} (1,M [p])\) for any motive \(M\). For smooth projective \(X\) over \(k\), by hard Lefschetz one will have a canonical decomposition \(R \Gamma_{\text{mot}} (X)\simeq \bigoplus H^ i_{\text{mot}} (X) [-i]\). The various classes of a \(d\)- codimensional algebraic cycle \(Z\) must come from a motivic class \(cl(Z)\in \operatorname{Hom} (1,R \Gamma_{\text{mot}} (X) [2d] (d))=: H^{2d}_{\text{abs}} (X, \mathbb{Q} (d))\) and the decomposition of \(R \Gamma_{\text{mot}} (X)\) thus leads to a series of classes \(cl_ n(Z)\in \text{Ext}^ n (1,H_{\text{mot}}^{2d-n} (X) (d))= H^ n_{\text{abs}} (H^{2d-n}_{\text{mot}} (X) (d))\). In Hodge theory one recovers the Hodge group and the intermediate jacobian, but higher Ext’s vanish, thus Hodge theory is not enough to get hold of all algebraic cycles.
Introducing algebraic \(K\)-theory, in particular the Adams groups \(K_ n(X)^{(j)}\) for a smooth variety \(X\) over a field, one conjectures that there are ‘Chern classes’ \[ ch^ j: K_ n(X)^{(j)} {\overset {\sim} {\longrightarrow}} H^{2j-n}_{\text{abs}} (X, \mathbb{Q}(j))= \operatorname{Hom}(1,R \Gamma(X) (j) [2j- n]). \] One enters Beilinson’s world!
The conjecture on the Chern classes has interesting consequences, e.g. for a pure motive \(M\) of weight \(w\) one will have \(\text{Ext}^ i (1,M)=0\) for \(i>-w\), and for an effective motive \(M\) of weight \(w\) and a positive integer \(b\) one has \(\text{Ext}^ i (1,M (w+b))=0\) for \(i>w+b\). Also, it would give weak Lefschetz theorems for Chow groups, and Murre’s conjectures on the Chow groups.
Of particular interest is the case \(X= \text{Spec} (F)\), where \(F\) is a number field. One would have \(\text{Ext}^ 1 (\mathbb{Q} (0), \mathbb{Q} (j))= K_{2j-1} (F)\otimes \mathbb{Q}\) and the higher Ext’s vanish. This imposes strong restrictions on the category of motives that are iterated extensions of Tate motives, and the systems of realizations of such motives. The paper closes with a program to obtain results for such systems of realizations.
For the entire collection see [Zbl 0788.00053].