Tate twists of Hodge structures arising from abelian varieties of type   IV
  We show that certain abelian varieties A have the property that for every Hodge structure V in the cohomology of A, every effective Tate twist of V occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV. 
Salman Abdulali
math.AG