Motives (mathematics)

Motives (mathematics)
Name	Motives
Field	Algebraic geometry
Introduced	1960s
Founders	Grothendieck, Weil, Serre
Related	Category theory, Homological algebra, Number theory

Motives (mathematics) are conjectural universal objects designed to capture the essential cohomological information of algebraic varieties in a single linear category. Originating in the work of Alexander Grothendieck and motivated by questions from André Weil and Jean-Pierre Serre, the theory aims to relate disparate cohomology theories via a category of motives that admits realisation functors to classical invariants.

History and Motivation

Grothendieck formulated the idea of motives in correspondence with André Weil and in the milieu of the Bourbaki seminars associated with Jean-Pierre Serre and Alexander Grothendieck, seeking to explain the parallels between Hodge theory exemplified by René Thom and Henri Poincaré, l-adic cohomology developed by Pierre Deligne addressing problems from the Weil conjectures, and algebraic cycles studied by Spencer Bloch and Yuri I. Manin. The foundational aspiration tied to conjectures propagated by Grothendieck, including the standard conjectures and the Bloch–Beilinson conjectures, connects to later work by Pierre Deligne on mixed Hodge structures, Vladimir Voevodsky on triangulated categories, and Beilinson on regulators, with interactions involving institutions such as the Institute for Advanced Study, École Normale Supérieure, IHÉS, Princeton University, and events like the Séminaire Bourbaki.

Classical and Pure Motives

The classical approach, championed in part by Grothendieck and manifested in the writings of Serge Lang and Jean-Pierre Serre, constructs pure motives from smooth projective varieties via correspondences and idempotent decompositions related to algebraic cycles studied by Spencer Bloch, James Milne, and Alexander Beilinson. Pure motives are categorized under equivalence relations such as numerical equivalence influenced by John Tate and Alexander Grothendieck, homological equivalence appearing in the work of Pierre Deligne, and rational equivalence following Samuel Lang and Oscar Zariski. The development of Chow motives and the theory of motives with coefficients drew on contributions from Endre Szemerédi-like combinatorial insights as well as explicit calculations by Yuri Manin and Jacob Murre, while comparisons to Hodge structures from Phillip Griffiths and Chris Peters and to l-adic representations investigated by Barry Mazur, Jean-Pierre Serre, and Richard Taylor illuminate the structure of pure motives.

Triangulated and Derived Motives

Triangulated and derived approaches, initiated by Vladimir Voevodsky and refined by Markus Rost, extend motives beyond the pure case to accommodate singular and nonprojective varieties, connecting to the motivic homotopy theory developed with Fabien Morel. Voevodsky's triangulated category of motives builds on techniques from Daniel Quillen's algebraic K-theory, Pierre Deligne's theories of weights, and the homotopical methods used by J. Peter May and Graeme Segal, integrating cycle complexes studied by Spencer Bloch and Andrei Suslin. Derived motives engage the formalism of triangulated categories from Alexandre Grothendieck's school and the enhancement frameworks studied by Maxim Kontsevich, Paul Seidel, and Bertrand Toen, and relate to stable homotopy categories explored at institutions like Harvard University and University of Chicago.

Realizations and Comparison Functors

Realization functors map motives to concrete cohomology theories, connecting to Hodge realizations by Pierre Deligne, l-adic realizations used in the proofs by Pierre Deligne of the Weil conjectures and later refinements by Richard Taylor, and de Rham realizations relevant to Alexander Grothendieck's crystalline cohomology and work by Arthur Ogus. The comparison isomorphisms between Betti cohomology associated with Phillip Griffiths' Hodge theory, l-adic cohomology examined by John Tate and Jean-Pierre Serre, and p-adic Hodge theory advanced by Jean-Marc Fontaine, Gerd Faltings, and Kiran Kedlaya are mediated by conjectural or constructed realization functors, with regulators and special values connected to Srinivasa Ramanujan-like identities and conjectures formulated by Don Zagier, A. N. Parshin, and Alexander Goncharov.

Applications and Connections

Motivic ideas permeate number theory through connections to the Birch and Swinnerton-Dyer conjecture influenced by Bryan Birch and Peter Swinnerton-Dyer, to the Langlands program articulated by Robert Langlands and pursued by Michael Harris and Laurent Lafforgue, and to modularity results by Andrew Wiles and Richard Taylor. In algebraic geometry, motives interact with moduli problems studied by David Mumford, David Gieseker, and Ravi Vakil; with derived categories examined by Alexander Kuznetsov and Dmitri Orlov; and with enumerative geometry and mirror symmetry investigated by Maxim Kontsevich, Cumrun Vafa, and Edward Witten. In homotopy theory, the influence of Daniel Quillen, J. P. May, and Graeme Segal appears via motivic homotopy categories, while interactions with mathematical physics surface in work by Edward Witten and Nikita Nekrasov. Institutions and collaborations across Princeton University, Stanford University, University of Cambridge, École Normale Supérieure, and the Clay Mathematics Institute have fostered advances and workshops.

Conjectures and Open Problems

Central conjectures shaping the field include Grothendieck's standard conjectures and their implications for Weil cohomology, the Bloch–Beilinson filtration conjecture advanced by Spencer Bloch and Alexander Beilinson, the Tate conjecture proposed by John Tate, and the Hodge conjecture formulated by W. V. D. Hodge and influenced by Phillip Griffiths. Other open problems connect to Fontaine–Mazur-type conjectures involving Jean-Marc Fontaine and Barry Mazur, the Beilinson conjectures on special values of L-functions influenced by Don Zagier and David Zagier, and challenges in constructing abelian categories of mixed motives articulated by Alexander Beilinson, Pierre Deligne, and Vladimir Voevodsky. Progress continues through work by Richard Taylor, Peter Scholze, Bhargav Bhatt, and Akshay Venkatesh, with active research environments at Princeton University, Harvard University, Oxford University, Institut des Hautes Études Scientifiques, and collaborative programs sponsored by the European Research Council.

Category:Algebraic geometry