Dense Algebraic Loci of Hodge Classes
We prove unconditionally that the Hodge conjecture is equivalent to two statements about abelian varieties, that the locus on which either statement holds is dense in every family and is either the whole family or a meagre set, that the real multiplications by $\sqrt6$ and by the cubic fields $\QQ(\zeta_{7})^{+}$ and $\QQ(\zeta_{9})^{+}$ are algebraic on explicit families of K3 surfaces, the cubic ones on the surfaces of Mumford type whose quaternion algebra is ramified at two real places, and that the Hodge conjecture holds on every power and every Hilbert scheme of points of the K3…
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