Motive (algebraic geometry)

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For other uses, see Motive.

In algebraic geometry, a motive (or sometimes motif) refers to 'some essential part of an algebraic variety'. Mathematically, the theory of motives is then the conjectural "universal" cohomology theory for such objects. In terms of category theory, it was intended to have a definition via splitting idempotents in a category of algebraic correspondences. The way ahead for that definition has been blocked for some decades, by the failure to prove the standard conjectures on algebraic cycles. This prevents the category from having 'enough' morphisms, as can currently be shown. While the category of motives was supposed to be the universal Weil cohomology much discussed in the years 1960-1970, that hope for it remains unfulfilled. On the other hand, by a different route, motivic cohomology now has a technically-adequate definition.

There is therefore no well-established theory of motives yet. Instead, we know some facts and relationships between them that (as generally accepted among mathematicians) point to the existence of general underlying framework. Some mathematicians prefer the word motif to motive for the singular, following French usage.

Contents

[edit] What is a motive?

[edit] Examples

Each algebraic variety X has a corresponding motive [X], so the simplest examples of motives are:

  • [point]
  • [projective line] = [point] + [line]
  • [projective plane] = [plane] + [line] + [point]

These 'equations' hold in many situations, namely:

Each motive is graded by degree (for example, the motive [X] is graded from 0 to 2 dim X). Unlike the usual varieties one can always extract each degree (as it is an image of the whole motive under some of projection). For example:

  • h = [elliptic curve] − [line] − [point]

is a 1-graded non-trivial motive.

[edit] The idea

The general idea is that one motive has the same structure in any reasonable cohomology theory with good formal properties; in particular, any Weil cohomology theory will have such properties. (The concept of a Weil cohomology was a working draft. Quite a number of candidates were proposed, for a cohomology theory with characteristic zero coefficients, applying to characteristic p geometry.)

There are many things one may be interested in for an algebraic variety, such as computing the number of rational points in some finite field. This information is already given by the Weil conjectures (which are now proven), and the standard conjectures are part of the effort to extend these results to characteristic 0.

Cohomology theories come with different structures:

One may ask whether there exists some universal theory which embodies all these structures and provides a common ground for equations like [projective line] = [line]+[point].

The answer is: people have tried to precisely define this theory for many years. The current name of this theory is the theory of motives.


[edit] Definition

The category of pure motives is constructed by the following formal procedure:

Consider a category of smooth projective schemes over some field k with correspondences as morphisms. The correspondences used must satisfy an adequate equivalence relation on algebraic cycles, ensuring that certain intersection-theoretic properties are preserved with respect to the Weil cohomology groups. Possible adequate equivalences are given by rational equivalence, algebraic equivalence, and homological equivalence, and numerical equivalence on cycles.

The resulting category has direct sums and tensor products, but is not abelian. Taking the Karoubi envelope of this category yields a pseudoabelian category which in particular adds all images of projectors; this is the category of effective motives. This category will contain a so-called Lefschetz motive whose tensor inverse, the Tate motive, is then formally adjoined to yield the category of pure motives.

The formal definition of a mixed motive is: [to be added]

[edit] Tannakian category approach

Through the technical machinery of Tannakian category theory (going back to Tannaka-Krein duality, but a purely algebraic theory), categories of motives are or should be equivalent to the category of linear representations of an algebraic group (or pro-algebraic group, if there is not a finite set of generating objects). An important application is to define a motivic Galois group; it is to the theory of motives what the Mumford-Tate group is to Hodge theory. Its purpose is to shed light on both the Hodge conjecture and the Tate conjecture, the outstanding questions in algebraic cycle theory. Again speaking in rough terms, the Hodge and Tate conjectures are types of invariant theory (the spaces that are morally the algebraic cycles are picked out by invariance under a group, if one sets up the correct definitions). The motivic Galois group has the surrounding representation theory. (What it is not, is a Galois group; however in terms of the Tate conjecture and Galois representations on étale cohomology, it predicts the image of the Galois group, or, more accurately, its Lie algebra.)

[edit] Remarks

Motives were part of the large-scale abstract algebraic geometry program initiated by Alexander Grothendieck. The consistency of a useful theory of motives still requires some conjectures to be proven and at present there are different definitions of motives. However, Grothendieck, with Bombieri, showed the depth of the motivic approach by producing a conditional proof of the Weil Conjectures, assuming the existence of such a theory of motives. The word 'motivic' occurring in the phrase motivic Galois group and elsewhere signifies a conceptual connection to the theory, but it must be accepted that the theory may not yet be in final form. See also motivic polylogarithm.

There is also a notion of a mixed motive: one expects that a mixed motive is to a motive as a mixed Hodge structure is to a Hodge structure.

[edit] External links

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