From: Bill Dubuque Newsgroups: sci.math.research Subject: Re: What does "motivic" mean? Date: 27 Aug 1998 05:19:15 -0400 greg@math.ucdavis.edu (Greg Kuperberg) writes: | | What does the word "motivic" mean? ... See Serre's nice introduction to motives, whose MR and Zbl are below. -Bill Dubuque ------------------------------------------------------------------------------ Serre, Jean-Pierre (F-CDF) Motifs. (French. English summary) [Motives] Journees Arithmetiques, 1989 (Luminy, 1989). Asterisque No. 198-200 (1991), 11, 333--349 (1992). MR 92m:14002 14A20 11G09 ------------------------------------------------------------------------------ This is a brief, nontechnical introduction into Grothendieck's theory of motives. The topics discussed by the author include cohomology theories for algebraic varieties, definitions of various categories of motives, motivic Galois groups and mixed motives. In the appendix, three short texts by A. Grothendieck about the "yoga of motives" are reproduced. Semisimplicity of the category of motives with respect to numerical equivalence, alluded to in the text, has been recently proved by 1150598U. Jannsen [Invent. Math. 107 (1992), no. 3, 447--452]. ------------------------------------------------------------------------------ Serre, Jean-Pierre Motifs. - Annexe: Quelques textes de Grothendieck sur les motifs. (Motives. - Appendix: Some texts by Grothendieck on motives). (French) [CA] Journees arithmetiques, Exp. Congr., Luminy/Fr. 1989, Asterisque 198-200, 333-349; Appendix: 342--347 (1991). [ISSN 0303-1179] Zbl. 759.14002 [For the entire collection see Zbl. 743.00058.] One of the best introductions to the theory of motives. In less then nine pages the background, motivation, (conjectural) formalism including basic examples and vistas to automorphic forms and mixed motives are explained in a most lucid way. As a special gift there is included an appendix consisting of 3 texts of A. Grothendieck on motives. Here both Serre's and Grothendieck's profound insights and their interplay reveal themselves once more. After a short introduction with a quotation of Grothendieck on the fascination of motives, the existence and compatibility of several cohomology theories is recalled. On the other hand, the apparent lack, in general, of compatibility isomorphisms between the various $\ell$-adic cohomology theories (for different values of the primes $\ell)$ is an unsatisfactory fact, and one feels that in many situations, e.g. when one has an algebraic correspondence between smooth projective varieties $X$ and $Y$ defined over some field $k$, morphisms of the kind $f$:$H\sp i\sb{\acute et}(X,\bbfQ\sb \ell)\to H\sp i\sb{\acute et}(Y,\bbfQ\sb \ell)$ are ``motivated'', and one would like to give meaning to the word ``motivated''. More precisely, one would like to be able to construct a $\bbfQ$-linear abelian category ${\cal M}(k)$ and a contravariant functor $h:{\cal V}(k)\to{\cal M}(k)$, where ${\cal V}(k)$ is the category of smooth projective $k$-varieties, having sufficiently nice properties to make $h(X)$ a universal rational cohomology in the sense that all other cohomologies of $X$ can be deduced from $h(X)$. This category ${\cal M}(k)$ is constructed, the essential ingredient being the definition of a morphism between two motives as an algebraic correspondence between the underlying varieties, considered modulo numerical equivalence. One might take other equivalence relations to obtain other theories of motives. No ${\cal M}(k)$ turns out to be a (graded) semi-simple Tannakian category, i.e. it has tensor products and internal Hom's with nice properties, relating it to the category of finite dimensional representations of some motivic Galois group, if one admits Grothendieck's standard conjectures, which have remained unproven. In case $k$ has characteristic zero, Deligne was able to construct a Tannakian category of motives by modifying the morphisms of motions, without relying on the standard conjectures. His correspondences are so-called absolute Hodge cycles. Deligne's motives have been shown to be successful in the theory of abelian varieties (over number fields). The standard examples of motives $h(X)$, $X=\bbfP\sb n$, $X$ the blown-up of a variety along a smooth closed subvariety, and $X$ a curve, are shortly discussed. Also, the motive of a cubic surface $X$ in $\bbfP\sb 3$ is given. It turns out to contain a piece $h\sp 2(X)=L\otimes(1\oplus V\sb 6)$, related to $\text{Pic}(X)$, where $L$ is the Lefschetz motive $h\sp 2(\bbfP\sb 1)$, and where $V\sb 6$ is an (Artin) motive of weight (degree) 0 and rank 6 coming from a Galois representation $\text{Gal}(\bar k/k)\to\text{GL}\sb 6(\bbfQ)$ with image contained in the Weyl group of the root system of type $E\sb 6$. This result is based on work of Yu. Manin. If $k$ admits an embedding into the complex numbers and if one accepts the standard conjectures as well as the Hodge conjecture, ${\cal M}(k)$ is semi-siple $\bbfQ$-linear Tannakian, and there is a fibre functor over $\bbfQ$, provided by Betti cohomology. The corresponding motivic Galois group $G\sb k$ is a pro-algebraic reductive $\bbfQ$-group. If $k$ is a number field the $G\sb k$ is related to the Serre groups $S\sb{\germ m}$. Taking the Tannakian subcategory ${\cal M}\sb X(k)$ generated by the motive $X$, one finds that the identity component $G\sp 0\sb {k,X}$ of its motivic Galois group $G\sb{k,X}$ is the Mumford-Tate group of $X$. In case $X=E$ is an elliptic curve without CM this leads to an explicit description of ${\cal M}\sb E(k)$. --- The relation between the category of motives over a number field and automorphic representations of reductive groups should lead to extremely interesting properties of $L$-functions within the ``Langlands philosophy''. The last section is concerned with the conjectural category of mixed motives. One should not restrict to smooth projective varieties, and this idea can be found already in Grothendieck's 1964 letter to Serre. The construction is far from clear. Deligne and Jannsen have candidates, but the final category of mixed motives remains mysterious. It should be related to algebraic $K$-theory via suitable extensions of pure motives. Also, for abelian varieties, similar extensions should shed much light on the Birch and Swinnerton-Dyer conjectures. The appendix consists of a letter of {\it A. Grothendieck}, dated 16/08/64, to Serre, and two extracts of ``Recoltes et Semailles''. It contains some of Grothendieck's reflections and insights on motives and related subjects such as fibre functors, the motivic Galois group and the standard conjectures. At several places, fundamental ideas, due to the author (toujours lui!), are mentioned. [ W.W.J.Hulsbergen (Breda) ] Citations: Zbl.743.00059 Keywords: motives; Tannakian category; Grothendieck standard conjectures; absolute Hodge cycles; mixed motives; Birch and Swinnerton-Dyer conjectures Classification: 14A20 Generalizations (algebraic spaces, etc.) 14C30 Transcendental methods 14C35 Appl. of methods of algebraic K-theory ============================================================================== From: egd@ams.org Newsgroups: sci.math.research Subject: Re: What does "motivic" mean? Date: Mon, 31 Aug 1998 19:49:49 GMT -----> Greg Kuperberg wrote: > > Subject: What does "motivic" mean? > Date: 23 Aug 1998 11:40:10 -0700 > From: greg@math.ucdavis.edu (Greg Kuperberg) > > What does the word "motivic" mean? That is, how do I tell a mathematical > object that is motivic from one which is not motivic? I have asked a > few slightly knowledgeable people this question, but the only responses > that I have gotten are discussions of what certain number theorists and > algebraic geometers are trying to do, and not any definitions of the > adjective "motivic" itself. > -- > /\ Greg Kuperberg (UC Davis) There is a "general interest" article on motivic cohomology at the address: http://www.ams.org/new-in-math/mathnews/motivic.html , mostly dealing with Voevodsky's work. - Edward Dunne -----== Posted via Deja News, The Leader in Internet Discussion ==----- http://www.dejanews.com/rg_mkgrp.xp Create Your Own Free Member Forum ============================================================================== From: greg@math.ucdavis.edu (Greg Kuperberg) Newsgroups: sci.math.research Subject: Follow-up: The meaning of "motivic" Date: 11 Nov 1998 17:11:25 -0800 A while ago I posted a query about the meaning of the word "motivic". I got many replies, and I should have followed up at the time, but I let it slip. The answer that was closest to the intent of my question came from Torsten Ekedahl. Briefly, there is a conjectured initial or universal cohomology theory for smooth, proper algebraic varieties called "motivic cohomology", sketched by Grothendieck. (Declaring that an algebraic variety is smooth and proper is analogous to declaring that a topological space is a closed manifold.) In order to make full sense as a cohomology theory, an imposing array of number-theoretic conjectures would have to hold; these are called Grothendieck's "standard conjectures". In general a mathematical construction is called "motivic" if it pertains to this conjectured theory. In some cases a "motivic foobar" has a more specific meaning, either out of tradition or because it is part of the theory of motives that is known to make sense. -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the xxx Math Archive Front at http://front.math.ucdavis.edu/ \/ * From A-hat to Z(G), ABC to ZFC *