Standard Conjectures on Algebraic Cycles

E680777

conjecture family conjecture in algebraic geometry mathematical conjecture

The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.

Try in SPARQL Jump to: Statements Referenced by

Statements (48)

Predicate	Object
instanceOf	conjecture family ⓘ conjecture in algebraic geometry ⓘ mathematical conjecture ⓘ
aim	provide foundational theory of algebraic cycles ⓘ relate algebraic cycles to cohomology ⓘ underpin theory of pure motives ⓘ
appliesTo	smooth projective varieties ⓘ varieties over arbitrary fields ⓘ
assumes	existence of Weil cohomology theories ⓘ
component	Hodge type standard conjecture ⓘ Künneth type standard conjecture ⓘ Lefschetz type standard conjecture ⓘ numerical equivalence equals homological equivalence conjecture ⓘ
concerns	algebraic cycles modulo homological equivalence ⓘ algebraic cycles modulo numerical equivalence ⓘ hard Lefschetz theorem for algebraic cycles NERFINISHED ⓘ positivity properties of intersection forms ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ motivic theory ⓘ
formulatedInDecade	1960s ⓘ
hasConsequence	semisimplicity of certain motive categories ⓘ standard properties of numerical equivalence ⓘ
implies	Lefschetz decomposition for algebraic cycles ⓘ Weil conjectures over finite fields ⓘ existence of certain algebraic correspondences ⓘ symmetry of Betti numbers for smooth projective varieties ⓘ
influenced	development of modern motive theory ⓘ research on algebraic K-theory ⓘ work on the Tate conjecture ⓘ
involves	Lefschetz operators NERFINISHED ⓘ algebraic correspondences ⓘ intersection theory ⓘ polarizations on cohomology ⓘ
mainTopic	algebraic cycles ⓘ cohomology of algebraic varieties ⓘ theory of motives ⓘ
motivation	construct a semisimple category of pure motives ⓘ explain properties of zeta functions of varieties ⓘ
namedAfter	algebraic cycles ⓘ
openProblemIn	algebraic geometry ⓘ number theory ⓘ
proposedBy	Alexander Grothendieck NERFINISHED ⓘ
relatedTo	Grothendieck motives NERFINISHED ⓘ Hodge conjecture NERFINISHED ⓘ Tate conjecture NERFINISHED ⓘ Weil cohomology theory NERFINISHED ⓘ
status	open ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hodge Conjecture → relatedTo → Standard Conjectures on Algebraic Cycles ⓘ