The standard conjectures on algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe standard conjectures describe algebraic cohomology classes on the product of a smooth projective variety with itself, for a fixed Weil cohomology theory: cohomology classes induced by an algebraic cycle with rational coefficients via the cycle class map. Grothendieck envisaged them as a route to proving that his construction of pure motives forms a semisimple abelian category, and pointed out that they also imply the hardest part of the Weil conjectures, the finite field analogue of the Riemann hypothesis, which was later proved unconditionally by other means by Pierre Deligne. 1 Proposed YearThe Tata Institute colloquium in Bombay where Grothendieck presented the conjectures was held in 1968; the paper itself was published in 1969, per the Wikipedia bibliography entry for Grothendieck (1969). Progress Toward ResolutionThe conjectures remain open in general. The case of curves holds immediately, Murre proved the case for surfaces in 1990, and Katz and Messing used the Weil conjectures to prove the case for smooth projective varieties over finite fields in any dimension in 1974. The two settings diverge sharply, as the live source states directly. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
Sources
1. Standard Conjectures on Algebraic Cycles (Wikipedia)
Wikimedia FoundationLead section
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
References list, Grothendieck 1969 entry
Grothendieck, A. (1969), "Standard Conjectures on Algebraic Cycles", Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968)
Status section, complex numbers versus finite fields
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.
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