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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.

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    $\begingroup$ Perhaps you should start looking at classical motives and the papers by Demazure, Manin, Kleiman, Scholl and others. $\endgroup$
    – Leo Alonso
    Commented Jun 11 at 11:47
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    $\begingroup$ It should also be noted that there are many (inequivalent) notions of motives, e.g. Chow motives, Nori motives, André motives, Voevodsky motives, etcetera. In all of these, every smooth projective variety $X$ has an associated motive, but $X$ is not 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 $n > 1$. $\endgroup$ Commented Jun 11 at 14:20
  • $\begingroup$ In a way, motives currently function like a philosophy rather than a theory, and each of the definitions used in the literature satisfies some of the desired properties, while other properties might be conjectural or false. There is no single unconditional theory of motives. $\endgroup$ Commented Jun 11 at 14:23
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    $\begingroup$ Actually, a semi-abelian variety A represents a homotopy invariant sheaf with transfers, and this gives a Voevodsky motive that is distinct from the motive of A. Yet I would rather suggest you to start from some "classical" motives; they are easier. Alternatively, you may try to read about 1-motives. $\endgroup$ Commented Jun 12 at 8:05
  • $\begingroup$ @MikhailBondarko Thank you for your comments. $\endgroup$ Commented Jun 12 at 11:29

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