Hodge conjecture
For most abelian varieties, the algebra Hdg*("X") is generated in degree one, so the Hodge conjecture holds. In particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension. However, constructed an example of an abelian variety where Hdg2("X") is not generated by products of divisor classes. generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic field, then Hdg2("X") is not generated by products of divisor classes. proved that in dimension less than 5, either Hdg*("X") is generated in degree one, or the variety has complex multiplication by an imaginary quadratic field. In the latter case, the Hodge conjecture is only known in special cases.