In algebraic geometry the idea of a motive intuitively 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.

What is a motive?

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:

• for de Rham cohomology and Betti cohomology over the complex numbers
• for étale cohomology (l-adic cohomology) over other fields whose characteristic is not l
• for the number of points over any finite field
• for the conductor (which is of course trivial for above)

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.

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.

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:

• Betti cohomology has the advantage of being defined over the integers
• de Rham cohomology comes with a mixed Hodge structure
• étale cohomology has a canonical Galois group action
• add here conductor and other stuff...

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.

uf. more explanation here

Definition

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

Consider a category (mathematics) of algebraic varieties 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 necessarily abelian. Taking the Karoubi envelope of this category yields a pseudoabelian category which in particular adds all images of projector (mathematics)s; 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]

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 the present moment there are different definitions 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.

Reading list

• math.QA/9904055: (http://xxx.lanl.gov/abs/math.QA/9904055) Maxim Kontsevich
  • Operads and motives in deformation quantization
• math.AG/9911179: (http://xxx.lanl.gov/abs/math.AG/9911179) A.Craw
  • An introduction to motivic integration
• (preprint, 1992): Alexander Beilinson, Pierre Deligne
  • Motivic polylogarithm and Zagier conjecture