A computation of a Chow motive: Guletskii and Pedrini
Some Open questions - a summary from a conference in Palo Alto
Grothendieck’s standard conjectures
Tate’s Conjecture, algebraic cycles and rational K-theory in characteristic p, by Thomas Geisser. We discuss conjectures on motives, algebraic cycles and K-theory of smooth projective varieties over finite fields. We give a characterization of Tate’s conjecture for varieties over finite fields in terms of motives and their Frobenius endomorphism and a criterion in terms of motives for rational and numerical equivalence over finite fields to agree. This together with Tate’s conjecture implies that higher rational K-groups of smooth projective varieties over finite fields vanish (Parshin’s conjecture). Parshin’s conjecture in turn implies a conjecture of Beilinson and Kahn giving bounds on rational K-groups of fields in finite characteristic. We go on to derive further corollaries.
arXiv: Experimental full text search
AG (Algebraic geometry)
Pure, Mixed
For mixed Tate motives, see Kahn
Ayoub has given a talk on motives of rigid varieties
Spitzweck: Operads, algebras and modules in model categories and motives. PhD thesis, 2001.
Thm: Let be additive and idempotent complete. If with the a finite set of orthogonal idempotents, then , with
MR1724886 (2001h:11083) Bloch, Spencer(1-CHI) Remarks on elliptic motives.
Roy Joshua - The Ohio State University Title: Motivic -algebras and the motivic dga Abstract: In this talk we will construct explicit -structures on the motives of smooth schemes, on the motivic complex and on complexes defining étale cohomology. This is contrasted with the construction of -structures on complexes defining singular homology and cohomology of complex algebraic varieties -several similarities and some surprising differences will emerge. Applications of such -structures include the construction of a category of mixed Tate motives for a large class of schemes over number fields and the construction of “classical” cohomology operations in motivic cohomology with finite coefficients.
André: Pour une théorie inconditionelle des motifs. Publ. Math. IHES, 1996.
Gillet and Soulé: Descent, motives and K-theory
André: Motifs de dimension finie. Bourbaki report, 2004.
André and Kahn: Construction inconditionnelle de groupes de Galois motiviques (2002)
André and Kahn: Nilpotence, radicaux et structure monoidales (2002)
Voevodsky preprint: Motives over simplicial schemes
Construction inconditionnelle de groupes de Galois motiviques, by Kahn and André
Gillet and Soulé: Descent, Motives and K-theory
J-invariant of linear algebraic groups , by Victor Petrov , Nikita Semenov , and Kirill Zainoulline, with nice applications to motivic decompositions. http://www.math.uiuc.edu/K-theory/0805
A note on relative duality for Voevodsky motives, by Luca Barbieri-Viale and Bruno Kahn: http://www.math.uiuc.edu/K-theory/0817
Pure motives, mixed motives and extensions of motives associated to singular surfaces, by J. Wildeshaus: http://www.math.uiuc.edu/K-theory/0856
Weight structures, weight filtrations, weight spectral sequences, and weight complexes (for motives and spectra) , by Mikhail V. Bondarko: http://www.math.uiuc.edu/K-theory/0843
On the multiplicities of a motive , by Bruno Kahn, also http://www.math.uiuc.edu/K-theory/0817
Motives and étale motives with finite coefficients , by Christian Haesemeyer and Jens Hornbostel: http://www.math.uiuc.edu/K-theory/0678
On the derived category of 1-motives, by Luca Barbieri-Viale and Bruno Kahn: http://www.math.uiuc.edu/K-theory/0851 (Aiming towards Deligne’s conjectures on 1-motives)
Geometric motives and the h-topology, by Jakob Scholbach: The main theorem of this paper is a reinterpretation of Voevodsky’s category of geometric motives: over a field of characteristic zero, with rational coefficients we obtain a description of the category avoiding correspondences by using the h-topology.
Deglise on generic motives
Some abstract stuff, implying that motives of one-dimensional schemes are Kimura finite-dimensional: Guletskii. This is also proved by other means by Mazza
The Geisser-Levine method revisited and algebraic cycles over a finite field, by Bruno Kahn: http://www.math.uiuc.edu/K-theory/0529
Kahn and Sujatha on birational motives
Semenov on a motivic decomposition
Regulators for Dirichlet motives: http://www.math.uiuc.edu/K-theory/0204
Goncharov on mixed elliptic motives: http://www.math.uiuc.edu/K-theory/0228. This is related to work of Wildeshaus in the 90s (elliptic analog of Zagier’s conjecture)
Here is something on 1-motives, also something here
Karpenko: construction of the Rost motive
Finite dimensional motives and the conjectures of Beilinson and Murre, by Vladimir Guletskii and Claudio Pedrini: http://www.math.uiuc.edu/K-theory/0617
Crystalline realizations of 1-motives, by F. Andreatta and L. Barbieri Viale: http://www.math.uiuc.edu/K-theory/0620
On the transcendental part of the motive of a surface , by Bruno Kahn , Jacob P. Murre , and Claudio Pedrini: http://www.math.uiuc.edu/K-theory/0759
Morava: On the motivic Thom isomorphism. The existence of a good theory of Thom isomorphisms in some rational category of mixed Tate motives would permit a nice interpolation between ideas of Kontsevich on deformation quantization, and ideas of Connes and Kreimer on a Galois theory of renormalization, mediated by Deligne’s ideas on motivic Galois groups.
Wildeshaus on the boundary motive. See also http://www.math.uiuc.edu/K-theory/0708
S. Nikolenko, N. Semenov, and K. Zainoulline
Chow motives of twisted flag varieties, by Baptiste Calmes, Viktor Petrov, Nikita Semenov, and Kirill Zainoulline: http://www.math.uiuc.edu/K-theory/0741
Réalisation l-adique des motifs triangulés géométriques, by Florian Ivorra
Intersection pairing and intersection motive of surfaces, by J. Wildeshaus
Levine’s motivic comparison theorem revisited, by Florian Ivorra
Shuji Saito [Shuji Saito1], Motives, algebraic cycles and Hodge theory (235–253)
Sermenev: Motif of an abelian variety (1974)
Remark: There might be a notion of motives for stacks. Ref Toen: On motives for DM stacks. Discusses Chow rings and Chow motives, 2 different defs. Motivation: Gromov-Witten invariants.
Some things including Andre and Bloch are found in folder AG/Motives
The two Motives volumes, edited by Jannsen, Kleiman and Serre. These contain for example: Deligne: A quoi servent les motifs? Scholl: Classical motives
Yves Andre: Introduction aux motifs.
For Absolute Hodge cycles: See LNM 900.
Manin: Correspondences, motives and monoidal transformations
M. Demazure, “Motives des variétés algébrique” , Sem. Bourbaki Exp. 365 , Lect. notes in math. , 180 , Springer (1971) pp. 19–38
S.L. Kleiman, “Motives” P. Holm (ed.) , Algebraic Geom. Proc. 5-th Nordic Summer School Math. Oslo, 1970 , Wolters-Noordhoff (1972) pp. 53–96
S.L. Kleiman, “Algebraic cycles and the Weil conjectures” A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , Dix exposés sur la cohomologie des schémas , North-Holland & Masson (1968) pp. 359–386
Saito: Motives, algebraic cycles and Hodge theory
Demazure: Motifs de varieties algebriques. Bourbaki report (1971)
Saavedra Rivano: Categories tannakiennes. LNM 265, 1972.
Milne: Motives over finite fields (in Motives vol)
This should be converted into a disambiguation page. Look at what comes up on a search and reorganise this introduction, with links to all the (key) types of motives, and a brief explanation of how they fit together. Maybe the same introduction will appear in the book project.
Types of motives: Artin motives, Artin-Tate motives (???), birational motives
For now we only give links to Pure motives and Mixed motives
The references and links below are in a chaos. Structure this when enough understanding.
See also Noncommutative motives
Barry Mazur: What is a Motive?.
Wikipedia article) on motives.
A very nice introduction by Bruno Kahn, in French.
Another nice introduction by Bloch (also exists in ps format)
A book review by Weibel (of Cycles, transfers and motivic homology theories)
A survey by Bruno Kahn, from the Handbook of K-theory.
Lectures of Kim and Sujatha at the Asian-French summer school, see also other references
Some slides by Kahn on pure motives. Also some slides on motivic Galois groups
A DG Guide to Voevodsky’s Motives, by A. Beilinson and V. Vologodsky: http://www.math.uiuc.edu/K-theory/0832
Clozel on automorphic objects corresponding to motives.
Lecture of Levine on Mixed motives and homotopy theory of schemes.
A book by Connes and Marcolli: Noncommutative geometry, quantum fields and motives.
Some nice things can be found towards the end of Jannsen‘s article on Deligne homology.
Voevodsky: Homology of schemes I. Discusses the moral of the notion of transfers in the introduction. One of the points seem to be that CTs are functorial wrt correspondences iff they are ordinary, i.e. come from coeffs in a complex of abelian sheaves. Another point is that the “motive” of a CW complex should be its stable homotopy type, if we by motive mean universal wrt all CTs, but if we work with ordinary theories, it should be its singular simplicial complex, as an object in the derived cat of abelian groups. More details provided.
Levine and Krishna explain some construction of homological motives, as the pseudo-abelian hull of a category , in which the objects are pairs of a smooth projective variety and an integer, and there is a functor sending this object to .
nLab page on Motives