What does "motivic" mean?
Greg Kuperberg
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)
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Torsten Ekedahl
> 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.
Many (most?) of the uses of "motivic" does not have a mathematically
precise definition. To clarify this let me start with the
corresponding noun; "motive". This notion was first introduced by
Grothendieck who wanted to express the idea of a universal cohomology
theory of (proper and smooth) algebraic varieties that would take its
values in an abelian category which in turn would have functors to the
categories in which the already known cohomology theories (étale, Betti
and de Rham cohomology) took their values mapping this universal
theory to the known ones. Grothendieck gave a definition of such a
category and associated to each algebraic variety an object in it (the
construction is fairly formal in the sense that it is constructed from
the category of algebraic varieties in a fairly formal
way).
Unfortunately, this is also where the trouble starts. In order to show
that the category he had constructed was abelian, Grothendieck had to
assume a collection of conjectures he formulated and which are known
under the name he gave them, the "standard conjectures". They however
are still very much beyond our reach and people have been somewhat
reluctant to exploit a theory that would build too much upon
them. There is another definition, introduced by Deligne (for
varieties in characteristic 0) using his idea of so called absolute
Hodge cycles but in order to have full confidence in it as the
"right" definition one would prefer to have available the "absolute
Hodge cycle conjecture".
Hence, the theory of motives lives somewhere in the shadow land
between mathematical philosophy and heuristics on the one side and
mathematics itself on the other. Sometimes it is quite enough to have
an object of Grothendieck's category and one does not need any
properties of the category itself and then one stays in the realm of
mathematics proper. When that is not enough the "theory of motives"
turns into a guiding philosophy rather than a well-defined
mathematical theory.
In many cases "motivic" simply means "pertaining to the theory of
motives" and thus should be accompanied with same caveats as the
theory itself. On the other hand sometimes it means something more
specific as for instance in the phrase "motivic cohomology" which are
well-defined abelian groups constructed out of Quillen's algebraic
K-theory (and are generalisations of the homomorphism groups of
Grothendieck's category).
I guess this does not answer your question :-)
Axel Schmitz-Tewes
could we say that motivic means in a vague way deduced from a 'motif'?.
So we could speak of a 'motivic' L-function as a generalisation of that
what can be deduced from an projective (or so.. ) algebraic variety
in contrast to L-functions like that of Dirichlet (and so ..).
would this clarify anything? :)
Axel
Torsten Ekedahl schrieb:
> ...
Torsten Ekedahl
> Hi,
>
> could we say that motivic means in a vague way deduced from a 'motif'?.
Yes, that was what I was trying to say below.
> So we could speak of a 'motivic' L-function as a generalisation of that
> what can be deduced from an projective (or so.. ) algebraic variety
> in contrast to L-functions like that of Dirichlet (and so ..).
Actually motivic L-functions belongs to the group of motivic objects that
make perfect mathematical sense. Whatever more sophisticated
properties the category of motives may or may not have their étale,
Betti and de Rham cohomology groups are defined and the L-function
only depends on these groups (seen as objects in their proper
categories). Well, except perhaps for some teeny-weeny problems at primes of
bad reduction. These motivic L-functions have Dirichlet L-functions as
very, very special cases.
Bill Dubuque
|
| 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
e...@ams.org
-----> Greg Kuperberg wrote:
>
> Subject: What does "motivic" mean?
> Date: 23 Aug 1998 11:40:10 -0700
> From: gr...@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
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