Interesting implications on the theory of motives if the Hodge conjecture holds

Question

For example,

Under the Hodge conjecture the Motivic galois group coincides with Mumford-Tate group.

The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as conjecture D (for singular cohomology) over fields of characteristic 0.

Answer 1

• If we assume the Hodge or the Tate conjecture, then the functor H∗MR is fully faithful on the category of Grothendieck motives (with homological or, under these assumptions equivalently, numerical equivalence).Hence it gives a linear algebra description of the conjectural abelian category of pure motives.

• Under the Hodge conjecture, the period conjecture can be reformulated in terms of Mumford-Tate group.

• The Hodge conjecture implies that the functor from the category of Grothendieck motives to the category of André motives and the functor from the category of André motives to the category of pure absolute hodge motives are equivalences of semi-simple Abelian categories.

Periods and Nori motives, Huber and Stach