Holmstrom Motives

Motives

A computation of a Chow motive: Guletskii and Pedrini


Motives

Some Open questions - a summary from a conference in Palo Alto

Standard conjectures

Grothendieck’s standard conjectures

Tate 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.

Bloch’s conjecture on surfaces?

Bloch/Murre/Beilinson filtration???

category: Open Problems


Motives

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

category: Search results


Motives

AG (Algebraic geometry)

category: World [private]


Motives

Pure, Mixed

category: Labels [private]


Motives

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 CC be additive and idempotent complete. If id X=Σe iid_X = \Sigma e_i with the e ie_i a finite set of orthogonal idempotents, then XX iX \cong \oplus X_i, with X i=Im(e i)X_i = Im(e_i)

MR1724886 (2001h:11083) Bloch, Spencer(1-CHI) Remarks on elliptic motives.


Roy Joshua - The Ohio State University Title: Motivic E E_{\infty}-algebras and the motivic dga Abstract: In this talk we will construct explicit E E_{\infty}-structures on the motives of smooth schemes, on the motivic complex and on complexes defining étale cohomology. This is contrasted with the construction of E E_{\infty}-structures on complexes defining singular homology and cohomology of complex algebraic varieties -several similarities and some surprising differences will emerge. Applications of such E E_{\infty}-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.


category: [Private] Notes


Motives

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)


Motives

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.

category: Definition


Motives

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.

de Jong’s list of references

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)

category: Paper References


Motives

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


Motives

Some notes by Grothendieck

Basic/introductory

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)

Barbieri-Viale

A book review by Weibel (of Cycles, transfers and motivic homology theories)

http://mathoverflow.net/questions/28803/what-is-the-proper-initiation-to-the-theory-of-motives-for-a-new-student-of-algeb

Arithmetic aspects

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

More advanced

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.


Motives

Writings of Grothendieck

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 Cor SCor_S, in which the objects are pairs (X,n)(X, n) of a smooth projective variety and an integer, and there is a functor CHCH sending this object to CH(X,n)CH(X, n).

nLab page on Motives

Created on June 10, 2014 at 21:14:54 by Andreas Holmström