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Related to Jim Milne's answer, one might mention that Deligne proved that for abelian varieties, all Hodge cycles are "absolutely Hodge" (i.e. when you think of them embedded diagonally inside the product of the algebraic de Rham cohomology and $\ell$-adic cohomology (for every $\ell$) and apply an automorphism of $\mathbb C$, the resulting cycles are again diagonally embedded rational cycles, and are in fact again Hodge). Note that if the Hodge conjecture holds, then this is certainly true (since the conjugate under any automorphism of $\mathbb C$ of an algebraic cycle is again an algebraic cycle).

On the one hand, this is much more than is known about the Hodge conjecture for more general classes of varieties.

On the other hand, one can't immediately extend this to other classes of varieties is because the motives of abelian varieties don't generate all motives over a field of char. 0 (in fact, far from it, as far as I know), a fact already brought up in Donu Arapura's answer.

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Related to Jim Milne's answer, one might mention that Deligne proved that for abelian varieties, all Hodge cycles are "absolutely Hodge" (i.e. when you think of them embedded diagonally inside the product of the algebraic de Rham cohomology and $\ell$-adic cohomology (for every $\ell$) and apply an automorphism of $\mathbb C$, the resulting cycles are again diagonally embedded rational cycles, and are in fact again Hodge). Note that if the Hodge conjecture holds, then this is certainly true (since the conjugate under any automorphism of $\mathbb C$ of an algebraic cycle is again an algebraic cycle).

On the one hand, this is much more than is known about the Hodge conjecture for more general classes of varieties.

On the other hand, one can't obviously immediately extend this to other classes of varieties is because the motive motives of abelian varieties don't generate all motives over a field of char. 0 (in fact, far from it, as far as I know), a fact already brought up in Donu Arapura's answer.

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Related to Jim Milne's answer, one might mention that Deligne proved that for abelian varieties, all Hodge cycles are "absolutely Hodge" (i.e. when you think of them embedded diagonally inside the product of the algebraic de Rham cohomology and $\ell$-adic cohomology (for every $\ell$) and apply an automorphism of $\mathbb C$, the resulting cycles are again diagonally embedded rational cycles, and are in fact again Hodge). Note that if the Hodge conjecture holds, then this is certainly true (since the conjugate under any automorphism of $\mathbb C$ of an algebraic cycle is again an algebraic cycle).

On the one hand, this is much more than is known about the Hodge conjecture for more general classes of varieties.

On the other hand, one can't obviously extend this to other classes of varieties is because the motive of abelian varieties don't generate all motives over a field of char. 0 (in fact, far from it, as far as I know), a fact already brought up in Donu Arapura's answer.