Les motifs - ou le coeur dans le coeur

It is with this fascinating title that A. Grothendieck presents in Recoltes et Semailles (cfr. Promenade à travers une oeuvre ou l'Enfant et la Mère) the subject of motives: the deepest of the twelve research themes around which he developed his "long-run" research program that literally revolutionized the field of algebraic geometry in the decade 1958-68. Motives were envisaged as the "heart of the heart" of the new geometry (arithmetic geometry) that Grothendieck invented following a scientific strategy based on the introduction of a series of new concepts organized on a progressive level of generality: starting with schemes, topos and sites then continuing with the yoga of motives and motivic Galois groups and finally introducing anabelian algebraic geometry and Galois-Teichmuller theory.
If the notions of scheme and topos were the two crucial ideas which constituted the original driving force in the development of this new geometry -- Grothendieck was evidently fascinated by the concepts of geometric point, space and symmetry -- it is only with the notion of a motive that one eventually captures the deepest structure, the heart of the profound identity between geometry and arithmetic.

Grothendienck wrote very little about motives. The foundations are documented in his unpublished manuscript "Motifs" and were discussed on a seminar at the Institut des Hautes Études Scientifiques, in 1967. We know, by reading Recoltes et Semailles, that he started thinking about motives in 1963-64. J.P. Serre has included in his paper "Motifs" an extract from a letter that Grothendieck wrote to him in August 1964 in which he talks (rather vaguely, in fact) of the notions of motive, fiber functor, motivic Galois group and weights.
Motives were introduced with the ultimate goal to supply an intrinsic explanation for the analogies occurring among the various cohomological theories for algebraic varieties: they were expected to play the role of a universal cohomological theory (the motivic cohomology) and also to furnish a linearization of the theory of algebraic varieties, by eventually providing (this was Grothendieck's viewpoint) the correct framework for a successful attack to the Weil's Conjectures on the zeta-function of an algebraic variety over a finite field.

Unlike in the framework of algebraic topology where the standard cohomological functor is uniquely characterized by the Eilenberg-Steenrod axioms in terms of the normalization associated to the value of the functor on a point, in algebraic geometry there is no suitable cohomological theory with integers coefficients, for varieties defined over a field k, unless one provides an embedding of k into the complex numbers. In fact, by means of such mapping one can form the topological space of the complex points of the original algebraic variety and finally compute the Betti (singular) cohomology. This construction however, does in general depend upon the choice of the embedding of k in the field of complex numbers. Moreover, Hodge cohomology, algebraic de-Rham cohomology, étale l-adic cohomology furnish several examples of different cohomology functors which can be simultaneously associated to a given algebraic variety, each of which supplying a relevant information on the topological space.

Grothendieck theorized that this plethora of different cohomological data should be somewhat encoded systematically within a unique and more general theory of cohomological nature that acts as an internal "liaison" between algebraic geometry and the collection of available cohomological theories. This is the idea of the "motif", namely the common reason behind this multitude of cohomological invariants which governs and controls systematically all the cohomological apparatus pertaining to an algebraic variety or more in general to ascheme.

The original construction of a category M of (pure) motives over a field k starts with two preliminary considerations. The first consideration is that M should be the target of a natural contravariant functor connecting the category C of smooth, projective algebraic varieties over a field k to M. Such functor should map an object X in C to its associated motive M(X). The second consideration is that this functor should, by construction, factor through any particular cohomological theory.

Now, keeping in mind this goal, one thinks about the axiomatizing process of a cohomological theory in algebraic geometry. This is done by introducing a contravariant functor X -> H(X) from C to a graded abelian category, where the sets of morphisms between its objects form K-vector spaces (K is a field of characteristic zero, that for simplicity, I fix here equal to the rationals). One also would like that any correspondence V--> W (an algebraic cycle in the cartesian product VxW that can be view as the graph of a multi-valued algebraic mapping) induces contravariantly, a mapping on cohomology and that the target category is suitably defined so that it contains among its objects any "Weil cohomological theory", namely a cohomology which satisfies among other axioms Poincaré duality and Künneth formula.

This preliminary disquisition helps one in formalizing the construction of the category of motives by following a three-steps procedure. One wishes to enlarges the category C in a precise way with the hope to produce also an abelian category. The three steps are shortly resumed as follows.

(1) One moves from C to a category with the same objects but where the sets of morphisms are the equivalence classes of rational correspondences. Here, the natural choice of the equivalence relation is the numerical equivalence relation as it is the coarsest one among the possible relations between algebraic cycles which can be seen to induce, via the cohomological axioms of any Weil cohomological theory, well-defined homomorphisms in cohomology.

(2) One enlarges the collection of objects of the category defined in (1), by formally adding kernels and images of projectors. This step is technically referred to as the "pseudo-abelian envelope" of the category defined in (1) and it is motivated by the expectation to define an abelian category of motives in which for instance, the Künneth formula can be applied.

(3) Finally, one considers the opposite of the category defined in (2).

Now, after having diligently applied all this abstract machinery, one would like to see a fruitful application of these ideas, in the form, for instance, of the proof of a major conjecture. However, one also perceives quite soon that a successful application of the yoga of motives is subordinated to a thorough knowledge of the theory of algebraic cycles, since the construction of the category M is centered on the idea of enlarging the sets of morphisms by implementing the notion of correspondence. It is for this reason that the Standard Conjectures (cohomological criteria for the existence of interesting algebraic cycles) were associated, since the beginning, to the theory of motives as they seem to play the "conditio sine qua non" a theory of motives has a concrete and successful application.
However, in order to put the Standard Conjectures in the right perspective and to avoid perhaps, an over-estimation of their importance, one should also record that Y. Manin gave in 1968, the first interesting application of these ideas on motives by producing an elegant proof of the Riemann-Weil hypothesis for non-singular three-dimensional projective unirational varieties over a finite field, without appealing to the Standard Conjectures. Moreover, we also know that the Weil's Conjectures have been proved by P. Deligne in 1974 without using neither the theory of motives nor the Standard Conjectures.

Almost forty years have passed since these ideas were informally discussed in the "Grothendieck's circle". An enlarged and in part still conjectural theory of mixed motives has in the meanwhile proved its usefulness in explaining conceptually, some intriguing phenomena arising in several areas of pure mathematics, such as Hodge theory, K-theory, algebraic cycles, polylogarithms, L-functions, Galois representations etc.

Very recently, some new applications of the theory of motives to number-theory and quantum field theory have been found or are about to be developed, with the support of techniques supplied by noncommutative geometry and the theory of operator algebras.
In number-theory, a conceptual understanding of the interpretation proposed by A. Connes of the Weil explicit formulae as a Lefschetz trace formula over the noncommutative space of adèle classes, requires the introduction of a generalized category of motives which is inclusive of spaces which are highly singular from a classical viewpoint. Several questions arise already when one considers special types of zero-dimensional noncommutative spaces, such as the space underlying the quantum statistical dynamical system defined by J.B. Bost and Connes in their paper "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking" (Selecta Math. (3) 1995). This space is a simplified version of the adèle classes and it encodes in its group of symmetries, the arithmetic of the maximal abelian extension of the rationals.

A new theory of endomotives (algebraic and analytic) has been recently developed in "Noncommutative geometry and motives: the thermodynamics of endomotives" (to appear in Advances in Mathematics). The objects of the category of endomotives are noncommutative spaces described by semigroup actions on projective limits of Artin motives (these are among the easiest examples of pure motives, as they are associated to zero-dimensional algebraic varieties). The morphisms in this new category generalize the notion of (algebraic) correspondences and are defined by means of étale groupoids to account for the presence of the semigroup actions.
An open and interesting problem is connected to the definition of a higher dimensional theory of noncommutative motives and in particular the set-up of a theory of noncommutative elliptic motives and modular forms.
A suitable generalization of the yoga of motives to noncommutative geometry has already produced some interesting results in the form, for example, of an analog in characteristic zero of the action of the Weil group on the étale cohomology of an algebraic variety.

It seems quite exciting to pursue these ideas further: the hope is that the motivic techniques, once suitably transferred in the framework of noncommutative geometry may supply useful tools and produce even more substantial applications than those obtained in the classical commutative context.

What is the "heart of the heart" of noncommutative geometry?