Motive (algebraic Geometry) - Conjectures Related To Motives

Conjectures Related To Motives

The standard conjectures were first formulated in terms of the interplay of algebraic cycles and Weil cohomology theories. The category of pure motives provides a categorical framework for these conjectures.

The standard conjectures are commonly considered to be very hard and are open in the general case. Grothendieck, with Bombieri, showed the depth of the motivic approach by producing a conditional (very short and elegant) proof of the Weil conjectures (which are proven by different means by Deligne), assuming the standard conjectures to hold.

For example, the Künneth standard conjecture, which states the existence of algebraic cycles πi ⊂ X × X inducing the canonical projectors H*(X) → Hi(X) ↣ H*(X) (for any Weil cohomology H) implies that every pure motive M decomposes in graded pieces of weight n: M = ⊕Gr_nM. The terminology weights comes from a similar decomposition of, say, de-Rham cohomology of smooth projective varieties, see Hodge theory.

Conjecture D, stating the concordance of numerical and homological equivalence, implies the equivalence of pure motives with respect to homological and numerical equivalence. (In particular the former category of motives would not depend on the choice of the Weil cohomology theory). Jannsen (1992) proved the following unconditional result: the category of (pure) motives over a field is abelian and semisimple if and only if the chosen equivalence relation is numerical equivalence.

The Hodge conjecture, may be neatly reformulated using motives: it holds iff the Hodge realization mapping any pure motive with rational coefficients (over a subfield k of C) to its Hodge structure is a full functor H : M(k)_Q → HS_Q (rational Hodge structures). Here pure motive means pure motive with respect to homological equivalence.

Similarly, the Tate conjecture is equivalent to: the so-called Tate realization, i.e. ℓ-adic cohomology is a faithful functor H: M(k)_Qℓ → Rep_ℓ(Gal(k)) (pure motives up to homological equivalence, continuous representations of the absolute Galois group of the base field k), which takes values in semi-simple representations. (The latter part is automatic in the case of the Hodge analogue).