Motives are objects that appear in algebraic geometry, which is closely related to algebraic cycles. It was considered by Grothendieck, and developed notably by Deligne, Voevodsky etc.

I have the following questions on them.

①Do we have a proof that abelian varieties and tori are motives? Do we have a theory on the category of objects consisting of motives and algebraic varieties?

②How abelian varieties and motives of abelian varieties are different even though abelian varieties themselves are motives? Why are abelian varieties called homology or cohomology of dimension 1?

③Motives are direct factors of cohomology, but I do not find any definition of motives using cohomology. I know their definition using hodge cycles and algebraic cycles. Are all motives obtained as direct factors of cohomology?

Thank you for your answers.

isnot a motive. You are probably thinking of Deligne's 1-motives, which is some ad hoc definition that tries to capture "$H^1$-phenomena" using abelian varieties, algebraic tori, and finitely generated free abelian groups. I am not aware of a generalisation of this notion to $n$-motives for $n > 1$. $\endgroup$