Standard conjectures on algebraic cycles

Standard conjectures on algebraic cycles
Name	Standard conjectures on algebraic cycles
Formulated	1960
Authors	Alexander Grothendieck
Field	Algebraic geometry
Status	Open

Standard conjectures on algebraic cycles The standard conjectures on algebraic cycles are a set of influential conjectures proposed by Alexander Grothendieck in the context of Weil conjectures and the development of a satisfactory cohomology theory for algebraic varietys over finite fields. They aim to relate algebraic cycles, intersection theory, and the behavior of Lefschetz operators in etale cohomology and other Weil cohomology theories, with deep connections to the work of figures such as Pierre Deligne, John Tate, and Jean-Pierre Serre.

History and formulation

Grothendieck formulated the standard conjectures in a series of letters and notes linked to the Weil conjectures and the search for a category of motives; these notes followed dialogues involving André Weil, Jean Leray, and colleagues at the Institute for Advanced Study. The conjectures were articulated to support Grothendieck’s vision of a motivic formalism that would underpin comparisons between de Rham cohomology, Betti cohomology, etale cohomology, and other Weil cohomology theories developed by researchers including Alexander Beilinson and Spencer Bloch. Grothendieck’s program influenced later developments such as the construction of mixed motives by contributors like Vladimir Voevodsky, Pierre Deligne, and Ulf Jannsen.

Statement of the conjectures

The standard conjectures are commonly enumerated as the Lefschetz type and the Hodge type (also called the Hodge standard conjecture), together with versions addressing numerical and homological equivalences, and positivity properties of algebraic cycles. In Grothendieck’s formulation the Lefschetz type conjecture asserts that the hard Lefschetz theorem analog for algebraic cycles holds in any Weil cohomology, relying on the action of a hyperplane class and its inverse given by a Lefschetz operator; the Hodge type conjecture posits a positivity statement for certain bilinear forms on primitive cohomology classes reminiscent of results from Hodge theory proven by W. V. D. Hodge and extended by Phillip Griffiths. Other statements compare equivalence relations: homological equivalence vs numerical equivalence on algebraic cycles, conjectures influenced by questions studied by John Tate and René Thom.

Relations to Weil conjectures and cohomology theories

Grothendieck introduced the standard conjectures to supply missing ingredients in proofs of the Weil conjectures and to give a motivic explanation for results obtained by Bernard Dwork and completed by Pierre Deligne. The Lefschetz type standard conjecture would provide an algebraic proof of the hard Lefschetz property used in Deligne’s proof of the last of the Weil conjectures, and the Hodge type conjecture would give the expected positivity for the Poincaré duality pairing in the context of etale cohomology and l-adic cohomology used by Deligne and by researchers such as Jean-Pierre Serre and Nicholas Katz. These conjectures also interact with theories like crystalline cohomology developed by Pierre Berthelot and with the theory of perverse sheaves as developed by Alexander Beilinson, Joseph Bernstein, and Deligne.

Known results and special cases

Progress on the standard conjectures has come through special cases and partial results. The Lefschetz type conjecture is known for projective spaces, abelian varieties (owing to work by Alfred Clebsch-era methods extended by Mumford and later by Lefschetz-inspired techniques), and for surfaces by techniques related to the Hodge index theorem proved by W. V. D. Hodge and generalizations by Igor Dolgachev and others. Results by Pierre Deligne on the Weil conjectures and by Alexander Grothendieck on motives have led to verification in cases involving K3 surfaces studied by PShinoda-style methods and later work by Shigeru Mukai and Duke of Edinburgh Prize-associated researchers (historical attribution varies), while coherent sheaf methods from Alexander Grothendieck and Jean-Pierre Serre underpin many special verifications. Advances in Hodge conjecture contexts by Claire Voisin and in Tate conjecture contexts by John Tate and Christopher Schoen contribute indirect evidence.

Consequences and applications

If proven, the standard conjectures would imply that numerical and homological equivalence coincide for algebraic cycles, yielding semisimplicity results for the category of pure motives envisioned by Grothendieck and influencing work by Yves André on motivated cycles. They would streamline relations between L-function properties studied by Andrew Wiles-adjacent communities, give algebraic foundations for the hard Lefschetz theorem in arbitrary Weil cohomology, and impact the study of automorphic forms where cohomological correspondences play a role in the Langlands program developed by Robert Langlands. The conjectures would provide tools for proving the standard conjectures’ expected consequences in intersection theory as formalized by William Fulton and in enumerative geometry related to problems pursued by Maxim Kontsevich and collaborators.

Counterexamples, obstructions, and open problems

No counterexample to the standard conjectures is known; nonetheless, obstructions arise from the difficulty of relating algebraic cycles to transcendental cohomology classes, as emphasized in comparative studies by Pierre Deligne, Phillip Griffiths, and Don Zagier. The Hodge standard conjecture is especially subtle in higher dimensions, intersecting unresolved issues in the Hodge conjecture and in the Tate conjecture where researchers like J. P. Serre and John Milnor have highlighted foundational obstacles. Open problems include verifying the conjectures for higher-dimensional varieties such as certain hyper-Kähler manifolds studied by Claire Voisin and D. Huybrechts, and establishing full compatibility with constructions in derived category approaches initiated by Alexander Bondal and Maxim Kontsevich. The conjectures remain central in contemporary research programs involving motivic Galois groups, l-adic representations studied by Richard Taylor, and the arithmetic geometry initiatives associated with institutions like the Institut des Hautes Études Scientifiques.

Category:Algebraic geometry