# What are motives used for?

*1994*

#### Translator’s note

*This page is a translation into English of the following:*

Deligne, P. “A quoi servent les motifs?” *Proc. Symp. in Pure Math.* **55** (1994), 143–161. publications.ias.edu/node/413

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `f36a214`

The first of the “standard conjectures” (Grothendieck [19], Kleiman [20]), the Lefschetz-style one, says that certain cohomology classes are algebraic.
Either way, if motives are the “direct factors” of algebraic varieties

Grothendieck tried to establish a catalogue of projective constructions of cycles, cf. [19].
For those for which we have calculated their cohomology class, the class can be expressed in terms of Chern classes of obvious vector bundles.
Even though I do not know of any counterexamples, it seems unlikely to me that the ring of cycle classes on

Instead of trying to construct cycles, we can try to construct vector bundles, and then take their Chern classes.
It is, in fact, like this that K. Kodaira and D.C. Spencer (1953) prove the Hodge conjecture for divisors (a theorem of Lefschetz), with the group

Unfortunately, in higher rank, we do not know how to construct vector bundles whose Chern classes are interesting any more than we know how to construct interesting cycles.

Over

The second: from the point of view of Hodge theory, the class of a cycle

- the cohomology class
c\in H_\mathbb{Z}^{d,d}(X) that we wish to be that of a cycleZ restricts to zero on theH_t ; - if
Z exists, then its cohomology classc determines the class of theZ_t in the intermediate JacobiansJ^d(H_t) ; and - the construction of
Z reduces to constructing the cyclesZ_t of classes defining a given section of the family of theJ^d(H_t) .

We can thus conclude that every element of

The aim of these notes is to show that, despite this lack of progress on the problem of constructing cycles, the philosophy of motives is a powerful tool.

I thank S. Bloch for his comments on a first version of these notes.

# 1 Motives

According to what we can and want to do, we have various definitions of motives available to us — or even none.
We need to distinguish pure motives, typically given by the cohomology of non-singular projective varieties, and mixed motives, where open and singular varieties are allowed.
The notion of a motive over

## 1.1

For a field

- Every algebraic variety
X onk should have motivic cohomology groupsH_\mathrm{mot}^i(X) , which are objects of{\mathscr{M}}(k) . In the pure case, we restrict to non-singular projective varietiesX , andH_\mathrm{mot}^i(X) should then be a pure motive of weighti . - For each of the usual cohomology theories
H , we should have a “realisation” functor\operatorname{real} , and isomorphismsH^i(X) = \operatorname{real}H_\mathrm{mot}^i(X). - There should be a tensor product
\otimes , with which the realisation functors are compatible.

## 1.2

The tensor product structure described above allows us to apply the theory of Tannakian categories, invented by Grothendieck to study the formalism of motives. References: Saavedra [28], Deligne–Milne [12], Deligne [15].

For *motivic Galois group* *fibre functor*

If we have an embedding of

In characteristic

The Tannakian theory is a *linear* analogue of a *set-theoretic* theory, namely that of the profinite *SGA 1: Revêtements étales et groupe fondamental*, Springer–Verlag, 1971, Lecture Notes in Math. **224**).
The analogy is as follows: if

## 1.3

In general, the motivic Galois group

Even though

The category *motivic affine group schemes* as being the opposite of the category of commutative Hopf algebras in

Here are three examples of such objects.

### 1.3.1

The motivic version of

### 1.3.2

The motivic Galois group: there exists a motivic version

### 1.3.3

Let

We can similarly define and work with motivic schemes, motivic torsors, … .

## 1.4

The abelianisation

In characteristic

For

## 1.5

One source of §1.1 (a) and (b) is the following example.
A smooth projective curve

We thus wish to be able to identify abelian varieties with certain motives of weight

Let

For

Again, we wish to be able to identify

## 1.6

Abelian varieties have spaces of modules, and these allow us to view certain quotients of symmetric Hermitian spaces by arithmetic groups in an algebraic way.
The case of motives is more complicated.
If

The variation

t inT'' ,m inF^i ,\nabla_t m inF^i : holomorphicity;t inT ,m inF^i ,\nabla_t m inF^{i-1} : transversality.

There exists a complex structure on the classifying space

The distribution

An analogous argument, also due to Griffiths, explains why the points of the intermediate Jacobian

## 1.7

When the distribution

Let

The data defining a Shimura variety *dual field*

A point of the Shimura variety over a field

- an exact
\otimes -functorx from the category\operatorname{Rep}(G) of representations ofG to the category of pure motives over\mathbb{F} ; and - an integer structure.
In terms of finite adelic realisations (the restriction of the product of
\ell -adic realisations), we can describe this as an isomorphism of\otimes -functorsx(V)_{\mathbb{A}^f} \xrightarrow{\sim} V\otimes\mathbb{A}^f, given up to composition with an element ofK .

The following condition should be satisfied.
For simplicity, we suppose that

Let

\iota be an embedding of\mathbb{F} into\mathbb{C} , extending the identity embedding ofE(G,X) into\mathbb{C} . There should existh\in X such that the following\otimes -functors from\operatorname{Rep}(G) to Hodge structures are isomorphic:(V,\rho)\mapsto V , endowed with the Hodge structure defined by\rho\circ h ;(V,\rho)\mapsto the Hodge realisation ofx(V) , after extending the base field to\mathbb{C} by\iota .

The fact that condition (c) is independent of the chosen complex embedding

- The de Rham realisation defines a fibre functor
V\mapsto x(V)_{\mathrm{DR}} on\operatorname{Rep}(G) that corresponds to aG -torsorP overF . The Hodge filtration of theX(V)_{\mathrm{DR}} is exact and compatible with the tensor product, and thus comes from a parabolicQ ofG^P and from\mu_{\mathrm{DR}}\colon\mathbb{G}_m\to Z(Q/\mathscr{R}_uQ) , whereZ denotes the centre, that lifts to a conjugation class of morphisms from\mathbb{G}_m toQ , Saavedra [28, IV, 2.4, p. 229]. Since\mathbb{F} containsE(G,X) , it makes sense to ask for the conjugation class corresponding to maps from\mathbb{G}_m toG to coincide with that ofh\circ\mu (forh\in X ). Condition (c) implies this. This explains the appearance of the dual field. - For a representation
(V,\rho) of weight0 :\rho\circ w trivial andV\otimes V\to\mathbb{Q} a symmetric invariant bilinear form such that, onV_\mathbb{R} ,B(v,h(i)w) is positive-definite and symmetric, we ask forx(V)\otimes x(V)\to 1 to be positive, for the desired polarisation (*loc. cit.*V, 2.4, p. 276) of the category of motives.

## 1.8

This motivic interpretation of Shimura varieties has been a guide for how to elaborate the axioms that characterise them, as well as for the determination of their conjugates (Borovoi [6]).

Let

# 2 Cohomology theories

## 2.1

A geometric construction, possible in one of the usual cohomology theories, should make sense “motivically,” and thus have an analogue in the other usual theories.

This principal has been crucial in developing mixed Hodge theory.
Grothendieck saw that each

To go any further, we must convince ourselves that every motive has a weight filtration

The same principle of transfer has allowed us to conjecture the asymptotic behaviour of a variation of Hodge structures on a punctured disc, or a product of punctured discs.

Let

We should not hope for an analogous behaviour for motives, since the transversality condition is non-trivial in Hodge theory.
In general, an object that describes the asymptotic behaviour should exist only on the punctured Zariski tangent space (assuming a semi-stable reduction), and it should not be enough to allow us to reconstruct the object we started with.
For example, if

The theory of Morihiko Saito ([29]), which gives the six operations (and the evanescent cycles) in mixed Hodge theory is in part inspired by the

It is again the philosophy of motives that led Grothendieck to conjecture the existence of the “mysterious” functor linking the

## 2.2

Let

I do not know in which sense

- For every usual cohomology theory
H , the corresponding realisation of\pi_1(X,x)_\mathrm{un} over\overline{k} has “local systems” (in the sense ofH ) onX as its linear representations, which are unipotent. - The functor “the fibre at
x ” gives an equivalence of categories:\begin{gathered} \text{smooth motives over }X_0\text{ that are iterated extensions} \\\text{of inverse images of motives over }k \\\downarrow \\\text{motives over }k\text{ endowed with an action of }\pi_1(X,x)_\mathrm{un}. \end{gathered}

In (a), the local systems in question are not assumed to be motivic. By (b), those that are motivic, however, form a faithful system of representations.

This philosophy has been inspired by the Hodge theory of

More generally, suppose that we have a set

a’. Over

b’. The functor “the fibre at

In (a’), the subquotients are taken over

In Hodge theory, this suggests the following questions.
Let

On

For ^{1}

The Hopf algebra

- the arrows in (2.2.1) are mixed Hodge morphisms; and
- the functor
M\mapsto M_x gives an equivalence between{\mathscr{A}} and the category of mixed Hodge structuresH with polarisable associated graded algebras that are endowed with a comodule structureH\to H\otimes{\mathscr{H}}(G) that is a mixed Hodge morphism.

# 3 Absolute cohomology

## 3.1

In each of the usual cohomology theories, the functors

We might hope that these functors

A triangulated category

In this section, we assume the existence of

For

For each of the usual cohomology theories, these motivic

## 3.2

If *absolute cohomology groups* of

These groups are often called the *motivic cohomology groups*.
They are vector spaces over

The

If

If

In fact, take

Analysing the proof, we can thus obtain, for every ample invertible sheaf

These arguments are essentially rational, since they depend on the difficult Lefschetz theorem.
If we had an integer variety of

## 3.3

Let

If

## 3.4

In Hodge theory,

## 3.5

Let

By a theorem of A. Weil [31, Theorem 7], the map

## 3.6

We now work, not over the spectrum of a field, but instead over a base

If

## 3.7

Suppose that

The example of the usual cohomology theories leads to us hope that the Chern classes

An optimistic conjecture is that the “Chern class” morphisms are isomorphisms:

This conjecture underlies the terminology “absolute motivic cohomology group” for the

Block [4] proposed an interpretation of the

## 3.8

Thanks to Beilinson, the conjecture (3.7.1) can be deduced from certain hypotheses, of which the most important are

- the functor
X\mapsto K_\bullet(X) factors throughR\Gamma_\mathrm{mot} ; and - the category
{\mathscr{D}}(k) is the derived category of the category of motives.

The conjecture has striking consequences for the two sides of (3.7.1).
They disappear, for trivial reasons, in different regions of the

- If
M is a motive, then\operatorname{Ext}^i(1,M)=0 fori<0 , and\operatorname{Ext}^0(1,M)=\operatorname{Hom}(1,M) . From this, it follows that\operatorname{Hom}(1,R\Gamma(X)(j)[N])=0 forN<0 , or forN=0 andj\neq0 . In K-theory, this gives the conjecture by Beilinson and Soulé thatK_n(X)^{(j)}=0 for2j<n , as well as for2j\leqslant n ifj\neq0 . - The Chow group
\operatorname{Ch}^j(X) of algebraic cycles of codimensionj is zero forj>\dim X , and, similarly,\operatorname{Ch}^j(X;n) is zero forj>\dim(X)+n . Furthermore,\operatorname{Ch}^j(X;n)=0 forn<0 .

Applying (3.7.1), we thus deduce:

—

- If
M is a pure motive of weightw , then\operatorname{Hom}^i(1,M)=0 fori>-w . - If
M is effective of weightw , andb>0 , then\operatorname{Hom}^i(1,M(w+b))=0 fori>w+b .

*Proof*. For (i), suppose that

For (ii), We again use the weak Lefschetz theorem (the

If (3.7.1) is an isomorphism then this would also imply the weak Lefschetz theorems for the Chow groups, as well as the conjectures on the Chow groups presented at this conference by J.P. Murre.

## 3.9

If

It is not clear to me if we should hope for an analogue of (3.9.1) over

A more concrete question is the following: suppose that we could define Galois cohomology classes in the

## 3.10

Let

The same circle of ideas allows us to construct a canonical central extension of

## 3.11

If

## 3.12

If

Here is a programme to prove the consequences of (3.12.1) for such realisation systems.

A. Define a Tannakian category

The idea is that Bloch’s higher Chow groups

A^{\bullet,m}=0 form<0 ;A^{\bullet,0} consisting only of\mathbb{Q} in degree0 ; andH^\bullet(A^{\bullet,n}) consisting only ofK_{2n-1}(F)\otimes\mathbb{Q} in degree1 .

We can then define

This part of the programme has essentially been carried out by May [26]. For a cubic variety, see the talk of Bloch in this volume.

B. Define, for each of the usual cohomology theories, a “realisation” fibre functor on the Tannakian category

C. Show that the mixed Tate motives in which we are interested (for example,

# Bibliography

*Topology*.

**1**(1962), 25–85.

*Current Problems in Mathematics*.

**24**(1984), 191–238.

*Astérique*.

**100**(1982).

*Adv. In Math.*

**61**(1986), 267–304.

*Algebraic cycles and higher K-theory, a correction*. 1989.

*Selecta Math. Soviet.*

**31**(1983-84), 3–59.

*Journées SMF de Dijon (1981)*.

**15**. Publ. Math. de l’Univ. de Paris VII, 1981: pp. 43–106.

*Inst. Hautes Études Sci. Publ. Math.*

**35**(1968), 107–126.

*Inst. Hautes Études Sci. Publ. Math.*

**44**(1974), 5–77.

*Proc. Sympos. Pure Math.*

**33**(1979), 247–290.

*Mémoires de La Soc. Math. France*.

**2**(1980), 23–33.

*Hodge Cycles, Motives, and Shimura Varieties*. Springer, 1982: pp. 9–100.

*Hodge Cycles, Motives, and Shimura Varieties*. Springer, 1982: pp. 261–279.

*Galois Groups over*\mathbb{Q} . MSRI Publications, 1989: pp. 79–297.

*Grothendieck Festschrift, Volume 2*. Birkhaüser, 1990: pp. 111–195.

*Hodge Cycles, Motives, and Shimura Varieties*. Springer, 1982: pp. 101–228.

*Amer. J. Math.*

**90**(1968), 805–865.

*Bull. Amer. Math. Soc.*

**76**(1970), 228–296.

*Bombay Coll. on Algebraic Geometry (1968)*. Oxford, 1969: pp. 193–199.

*Dix Exposés Sur La Théorie Des Schémas*. Adv. Stud. Pure Math., 1968: pp. 359–386.

*Proc. Nat. Acad. Sci.*

**39**(1953), 872–877.

*An addendum to “Notes towards Block–Deligne motives”*. 1991.

*Crelle*.

**378**(1987), 113–220.

*Selected Papers*. Chelsea, 1924.

*Ann. Sci. École Norm. Sup.*

**2**(1969), 1–62.

*Notes toward Bloch–Deligne motives*. 1991.

*Izv. Akad. Sci. URSS*.

**46**(1968), 1011–1046.

*Catégories tannakiennes*. Springer–Verlag, 1972.

*Lecture Notes in Math.*

**265**.

*Res. Inst. Math. Sci*.

**62**(1986), 360–363.

*Invent. Math.*

**80**(1985), 489-542 and 543-565.

*Math. Ann*.

**128**(1954), 95–127.

*Comp. Math*.

**34**(1977), 199–209.

*[Trans.] In the original, this equation is labelled (2.2.21), but it seems like it should instead be (2.2.1).*↩︎