Number/Representation Theory

Grothendieck envisioned that proving his standard conjectures would lead to the construction of a Tannakian category of motives, but the standard conjectures are still open in general. One of them, the Hodge standard conjecture, holds in characteristic 0 due to Hodge theory, and another one, the Kunneth type standard conjecture, follows in characteristic p from the Riemann hypothesis part of the Weil conjectures. We will combine these notions to unconditionally construct a Tannakian category of "motives for almost-algebraic cycles" in characteristic 0 that agrees with Grothendieck's construction if the Tate and Hodge conjectures hold.