Grothendieck's standard conjectures

Grothendieck's standard conjectures
Name	Grothendieck's standard conjectures
Field	Algebraic geometry
Proposer	Alexander Grothendieck
Year	1960s

Grothendieck's standard conjectures are a set of conjectures formulated in the 1960s by Alexander Grothendieck proposing deep properties of algebraic cycles and their relation to cohomology theories. They aim to explain the formal similarities between Weil conjectures proved by Pierre Deligne and expected properties of motives sought by Grothendieck, linking structures in Paris, École Normale Supérieure, IHÉS, and broader developments in algebraic geometry. The conjectures remain open in general and have guided research intersecting work of Alexander Grothendieck, Pierre Deligne, Jean-Pierre Serre, John Tate, and others.

Introduction

Grothendieck introduced these conjectures while developing the program of motives and the foundations of étale cohomology at institutions such as IHÉS and Université Paris-Sud. The conjectures were motivated by phenomena observed in the proofs of the Weil conjectures by Pierre Deligne and by the emerging theories of Hodge theory at Institut des Hautes Études Scientifiques and Collège de France. They assert statements about algebraic cycles on smooth projective varieties over fields like finite field, number field, and complex numbers, connecting to work of André Weil, Alexander Grothendieck, Jean-Pierre Serre, and John Tate.

Statement of the Conjectures

The standard conjectures include several interrelated assertions, traditionally named the Lefschetz-type and Hodge-type conjectures. One central claim is that for a smooth projective variety the Lefschetz operator yields algebraic correspondences, linking to formalism used by Pierre Deligne and Alexander Grothendieck in the theory of Weil cohomology and cycle class maps. Another asserts positivity properties of certain pairings on algebraic cycles, paralleling statements from Hodge theory familiar to researchers at Université de Paris and Princeton University. The conjectures are formulated to ensure that numerical and homological equivalence coincide in the context envisioned by John Tate and that the category of pure motives admits a semisimple tannakian description similar to frameworks developed at Institute for Advanced Study.

Motivations and Consequences

Grothendieck proposed these conjectures to reconcile results from the proofs of the Weil conjectures by Pierre Deligne with the desired properties of a universal cohomology theory advocated by Alexander Grothendieck and collaborators at IHÉS and Bourbaki. If true, the conjectures would imply the Künneth projectors are algebraic, provide a path to the Tate conjecture over finite fields, and establish semisimplicity for categories of motives as envisaged by Grothendieck and Serre. Consequences would touch work by John Tate, Pierre Deligne, Alexander Beilinson, and others developing the arithmetic of L-functions and structures at Harvard University and Princeton University.

Known Results and Partial Progress

Partial results include proofs in special situations: for divisors on surfaces following techniques related to Hodge theory developed by Phillip Griffiths and in the case of abelian varieties via methods associated with Lefschetz theorems and results of Tate and Weil. The conjectures are known for varieties admitting certain decompositions, for cases treated by Pierre Deligne in low-dimensional settings, and for specific classes such as varieties with cellular decompositions studied by researchers at University of Cambridge and Stanford University. Work by Yves André, Uwe Jannsen, J. S. Milne, and others has clarified implications and established equivalences between variants in the presence of standard hypotheses common in literature from École Polytechnique and University of Bonn.

Relations to Other Conjectures and Theories

The standard conjectures are intimately related to the Tate conjecture, the Hodge conjecture, and the general theory of motives initiated by Alexander Grothendieck and furthered by Pierre Deligne and Saavedra Rivano. They are expected to imply semisimplicity results central to tannakian formalism as developed by Saavedra Rivano and applied in work by Deligne at IHÉS. Connections also reach the study of L-functions as pursued at institutions like Princeton University and Harvard University, and they influence approaches to categorical frameworks explored at Institut des Hautes Études Scientifiques and University of Chicago.

Examples and Special Cases

Known verifications include surfaces and abelian varieties where results of Lefschetz, Tate, and Weil provide the necessary algebraicity of correspondences. Cellular varieties and certain low-dimensional families analyzed by Pierre Deligne and collaborators serve as explicit examples. Counterparts in complex geometry relate to classical results in Hodge theory developed by Carl Ludwig Siegel and elaborated by Phillip Griffiths and Wilhelm P. Spaulding in low-dimensional special cases studied at Harvard University and Princeton University.

Category:Algebraic geometry