Standard conjectures on algebraic cycles

In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his construction of pure motives gave an abelian category that is semisimple. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely Weil's Riemann hypothesis (i.e. an analog over finite fields of the well known Riemann hypothesis) that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see Kleiman (1968). The standard conjectures remain open problems, so that their application gives only conditional proofs of results. In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.

The classical formulations of the standard conjectures involve a fixed Weil cohomology theory H. All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on the cohomology of a smooth projective variety

H^∗(X) → H^∗(X)

induced by an algebraic cycle with rational coefficients on the product X × X via the cycle class map, which is part of the structure of a Weil cohomology theory.

Conjecture A is equivalent to Conjecture B (see Grothendieck (1969), p. 196), and so is not listed.

The situations over the field of complex numbers and over finite fields are completely different. Hodge standard conjecture is true over the field of complex numbers, but largely unknown over finite fields. On the other hand, Standard conjecture C is known over finite fields, but largely unknown over complex numbers.