Motives---Grothendieck's Dream. J.S. Milne   Top
Expository Notes
A Primer of Commutative Algebra
Motives---Grothendieck's Dream
What is a Shimura Variety?
Introduction to Shimura Varieties
Shimura Varieties and Moduli
Tannakian Categories
The Work of Tate
Errata
pdf (current version 2.04).

Abstract

Grothendieck introduced the notion of a "motif" in a letter to Serre in 1964. Later he wrote that, among the objects he had been privileged to discover, they were the most charged with mystery and formed perhaps the most powerful instrument of discovery. In this article, I shall explain what motives are, and why Grothendieck valued them so highly.

Contents

  1. Cohomology in topology
  2. Cohomology in algebraic geometry
  3. Why is there no algebraically defined Q cohomology?
  4. Algebraic cycles
  5. Definition of motives
  6. What is known about the category of motives
  7. The Weil conjectures revisited
  8. Zeta functions of motives
  9. The conjecture of Birch and Swinnerton-Dyer
  10. Final note

About

The origin of this article is a "popular" lecture, What is a Motive?, that I gave at the University of Michigan on February 3, 2009. A Chinese translation was published in Mathematical Advances in Translation, Vol.28, No.3, 193-206, 2009 (Institute of Mathematics, Chinese Academy of Sciences) (pdf; translation by Xu Kejian, Qingdao University). It is also to be published in English in Open Problems and Surveys of Contemporary Mathematics, SMM 6, pp. 325-342 (2013?).

History

v1.00 (Feb 18, 2009). First version on the web.
v1.02 (March 9, 2009; 12 pages). Minor fixes; redrew picture with jgpfdraw. pdf (version 1.02).
v2.00 (April 26, 2009; 16 pages). Revised; expanded.
v2.01 (May 7, 2009; 16 pages). Fixed misprints.
v2.02 (May 17, 2009; 17 pages). Changed title; added more references to the literature.
v2.03 (June 7, 2009; 17 pages). Minor expository changes. pdf (version 2.03).
v2.04 (April 24, 2012; 15 pages). Revised for publication (in English).

Deligne on the word "yoga"

In mathematics, there are not only theorems. There are, what we call, "philosophies" or "yogas," which remain vague. Sometimes we can guess the flavor of what should be true but cannot make a precise statement. When I want to understand a problem, I first need to have a panorama of what is around it. A philosophy creates a panorama where you can put things in place and understand that if you do something here, you can make progress somewhere else. That is how things begin to fit together.
In Mathematicians, Mariana Cook, PUP, 2009, p156.

On the word "motive"

In English there are two words.
motif: A distinctive, significant, or dominant idea or theme; Art. a distinctive feature, subject, or structural principle in a composition or design; in literature or folklore, a particular or recurrent event, situation, theme, character, etc.... . [This word is sometimes written "motive".]

motive: A factor or circumstance inducing a person to act in a certain way....

(Shorter Oxford Dictionary)
Both are borrowed from the French word "motif" which can have both meanings. Grothendieck obviously means motif with the first meaning.

When I was preparing SLN900 for publication, both "motif" and "motive" were in use for Grothendieck's notion, and I had to choose between them. I chose the anglicized form, partly because I'd come across people (Bernstein, Donington) who use "motive" for "motif" in the musical sense. Since then, "motive" has predominated.

In his 1968 article (in Russian), Manin explains the word by quoting Herbert Read (an English poet and critic):

Cézanne's method of painting was first to choose his 'motif' --- a landscape, a person to be portrayed, a still-life; then to bring into being his visual apprehension of this motif; and in this process to lose nothing of the vital intensity that the motif possessed in its actual existence. (A Concise History of Modern Painting).
In his review (BAMS 39 (2002), p138), Weibel paraphrases Read. As far as I know, Grothendieck was simply using a fairly common French word "motif", and Weibel's claim (ibid.) that he borrowed it from Cézanne is wrong.

More from Herbert Read:

Cézanne's immediate predecessors, the Impressionists, had seen the world subjectively---that is to say, as it presented itself to the senses, and for each occasion there must necessarily be a separate work of art. But Cézanne wished to exclude this shimmering and ambiguous surface of things and penetrate to the reality that did not change, that was present beneath the bright but deceptive picture presented by the kaleidoscope of the senses.

For a comparison of Grothendieck's and Cézanne's notions of "motif" by Xu Kejian, see here (Chinese).