 Then the Tate conjecture for divisors on "X" is equivalent to the Birch-Swinnerton-Dyer conjecture for the Jacobian variety of "F".By contrast, the Hodge conjecture for divisors on any smooth complex projective variety is known (the Lefschetz (1,1)-theorem).Probably the most important known case is that the Tate conjecture is true for divisors on abelian varieties.