Each algebraic variety X has corresponding motive [X], so the simplest examples of motives are:
- [projective line] = [point] + [line]
- [projective plane] = [plane] + [line] + [point]
These 'equations' hold in many situations, namely:
- for de Rham cohomology and Betti cohomology over complex numbers
- for étale cohomology over other fields
- for number of points over any finite field
- for conducttor (which is of course trivial for above)
Each motive is graded by degree (for example motive [X] is graded from 0 to 2 dim X). Unlikely to usual varieties one can always extract each degree (as it's an image of the whole motive under some of projection), example:
- h = [elliptic curve] - [line] - [point]
is 1-graded non-trivial motive.
The general idea is that one motive can be realized in different cohomology theories,
There are many things one may be interested in for an algebraic variety. It's interesting to compute its number of points in some finite field. Its cohomology come with different structures:
- Betti cohomology have the advantage of being defined over integer numbers
- de Rham cohomology come with mixed Hodge structure
- étale cohomology have canonical Galois group action
- add here conductor and other stuff...
One may ask whether there exists s universal theory which embodies all these structures and provides common ground for equations like [projective line] = [line]+[point].
The answer is: people try to precisely define this theory for many years. The current name of this theory is theory of motives.
uf. more explanation here
Motives are part of large abstract algebraic geometry program started by Alexander Grothendieck. The consistency of theory of motives still reqires some conjectures to be proven and at the present monent there are different definitions of motives.