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Classical motives. (English) Zbl 0814.14001

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 1, 163-187 (1994).
Here the reader will find an elegant approach to the theory of classical motives, i.e. (sums of) pure motives, with some emphasis on geometric aspects. In particular, special attention is payed to Chow motives, i.e. one starts from smooth projective varieties over some field and takes for the morphisms the cycle groups on products of such varieties modulo rational equivalence, which is the finest adequate equivalence relation for algebraic cycles. The coarsest adequate equivalence is numerical equivalence. The corresponding motives are called Grothendieck motives. Fixing some Weil cohomology one may speak of homological equivalence which is a priori finer than numerical equivalence but coarser than rational equivalence. It was conjectured by Grothendieck that, however, homological and numerical equivalence coincide, thus implying that the theory of motives can be considered as a universal cohomology theory. The paper consists of an introduction, six sections and references.
The first section deals with the definition and formal properties of motives for any adequate equivalence relation on cycles. A motive is defined as a triple \((X,p,m)\), where \(X\) is a smooth projective variety over some field \(k\) assumed to be purely \(d\)-dimensional, \(m\) is an integer, and \(p\) is a projector, i.e. an idempotent in the space of self- correspondence of \(X\) of degree zero \(\text{Corr}^ 0(X,X)\) which is just \(A^ d (X \times X) = {\mathcal Z}^ d (X \times X) \otimes \mathbb{Q}/\sim\) with \(\sim\) any of the aforementioned adequate equivalence relations. Motives form a category \({\mathcal M}_ k\) with morphisms given by \(\text{Hom}_{{\mathcal M}_ k}((X,p,m), (Y,q,n)) = q \circ \text{Corr}^{n-m}(X,Y) \circ p = q \circ A^{d+n-m} (X \times Y) \circ p\). There are two distinguished motives: \(\mathbf{1} = (\text{Spec} k,\text{id},0)\) (the unit motive) and \(\mathbb{L} = (\text{Spec }k,\text{id},- 1)\) (the Lefschetz motive). One defines the direct sum \((X,p,m) \oplus (Y,q,m) = (X\coprod Y,p \oplus q,m)\). For a smooth projective variety \(X\) one defines its associated motive by \(h(X) = (X,\text{id},0)\), where \(\text{id}\) is the class of the diagonal \(\Delta_ X\) in \(A^ d(X \times X)\). E.g. for \(X = \mathbb{P}^ 1\), \(\Delta_ X\) is rationally equivalent to the sum of \(\{x\} \times \mathbb{P}^ 1\) and \(\mathbb{P}^ 1 \times \{x\}\) for any \(x \in \mathbb{P}(k)\), and this gives a decomposition \(\text{id}_{\mathbb{P}^ 1} = p_ 0 + p_ 2\), \(p_ 2 = {}^ t p_ 0\), and a canonical decomposition \(h(\mathbb{P}^ 1) = h^ 0(\mathbb{P}^ 1)\oplus h^ 2 (\mathbb{P}^ 1) = \mathbf{1} \oplus \mathbb{L}\). More generally, \(h(\mathbb{P}^ n) = \mathbf{1} \oplus \mathbb{L} \oplus \cdots \oplus \mathbb{L}^ n\), where \(\mathbb{L}^ i\) means \(\mathbb{L}^{\otimes i}\), \(i \in \mathbb{N}\). One can define the tensor product of two motives by \((X,p,m) \otimes (Y,q,n) = (X \times Y, p \otimes q, m+n)\). The diagonal defines a product structure on \(h(X)\): \(h(X) \otimes h(X) = h(X\times X)\overset {\Delta^*} \longrightarrow h(X)\). One has \((X,p,m) = (X,p,0) \otimes \mathbb{L}^{-m}\), and any motive is a direct factor of some \(h(X) \otimes \mathbb{L}^ n\). This makes it possible to define the direct sum of any two motives \((X,p,m)\) and \((Y,q,n) = (X \times (\mathbb{P}^ 1)^{n-m} \coprod Y, p' \oplus q,n)\), where \(p'\) is a projector of \(X \times (\mathbb{P}^ 1)^{n-m}\) constructed by means of \(p\) and the canonical projector \(p_ 2\) of \(\mathbb{P}^ 1\). One also has a dual motive \((X,p,m)^ \vee = (X,{}^ tp, d - m)\), in particular \(h(X)^ \vee = h(X) \otimes \mathbb{L}^{-d}\) (‘Poincaré duality’). For arbitrary motives \(M\), \(N\) and \(P\) one obtains the formula for the Hom’s, \(\text{Hom} (M \otimes N,P) = \text{Hom} (M,N^ \vee \otimes P)\) and thus one may define the internal Hom, \(\text{Hom} (M,N):= M^ \vee \otimes N\). Altogether, \({\mathcal M}_ k\) becomes a rigid, pseudo-abelian, \(\mathbb{Q}\)- linear tensor category. Jannsen proved that \({\mathcal M}_ k\) is abelian semi-simple if and only if one takes numerical equivalence in the \(A^*(X \times Y)\). Thus with Grothendieck’s conjecture one would have that \({\mathcal M}^{\text{num}}_ k = {\mathcal M}^{\text{hom}}_ k\) is a semi-simple \(\mathbb{Q}\)-linear tannakian category (after suitable definition of commutativity and associativity constraints). On the other hand it is shown in the third section that \({\mathcal M}^{\text{rat}}_ k\) is not abelian in general.
One can recover the cycle class groups \(A^*(X)\) by the formula \(A^ i(X) = \text{Hom}({\mathbb{L}}^ i,h(X))\) which follows from the definition of motives. For an arbitrary motive \(M\) one defines \(A^ i(M) := \text{Hom}({\mathbb{L}}^ i,M)\), and one obtains a contravariant functor from the category of smooth projective varieties over \(k\) to finite dimensional \(\mathbb{Q}\)-vector spaces \(\omega_ M(Y) = A^*(M \otimes h(Y))\). Manin’s identity principle says that a morphism \(f : M \to N\) in \({\mathcal M}_ k\) is an isomorphism iff the induced map \(\omega_ f(Y) : A^* (M\otimes h(Y)) \to A^* (N \otimes h(Y))\) is an isomorphism for all smooth projective varieties \(Y\). Two morphisms \(f,g : M \to N\) coincide iff \(\omega_ f(Y) = \omega_ g(Y)\) for all \(Y\). Also, a sequence \[ 0 \rightarrow M' \to M \to M'' \to 0 \] in \({\mathcal M}_ k\) is exact iff the induced sequence \[ 0 \to \omega_{M'}(Y) \to \omega_ M(Y) \to \omega_{M''}(Y) \to 0 \] is exact for all \(Y\). As applications the calculation of the motive of a projective bundle and of a blow-up are given. This last result was used in the sixties by Manin to prove the Weil conjectures for three-dimensional unirational varieties over finite fields.
In the third section the relation between the motive of a curve and its Jacobian, due in essence to Weil, is studied. If \(X\) and \(X'\) are smooth projective curves with Jacobians \(J\) and \(J'\), respectively, one has \(\text{Hom}(h^ 1(X), h^ 1(X')) = \text{Hom} (J,J') \otimes \mathbb{Q}\), where one uses the decomposition \(h(X) = h^ 0(X) \oplus h^ 1(X) \oplus h^ 2(X)\) and similarly for \(X'\), where \(h^ 0\) and by transposition \(h^ 2\) are determined by projectors \(p_ 0\) and \(p_ 2\) defined by some zero cycle of positive degree (e.g. a rational point if there are) on \(X\) and then \(h^ 1(X)\) is just \((X,p_ 1,0)\) with \(p_ 1 = \text{id} - p_ 0 - p_ 2 \in \text{Corr}^ 0(X, \overline {X}) = A^ 1 (X \times X)\), and similarly for \(X'\). Furthermore, \[ \text{Hom}(\mathbb{L},h^ 1 (X)) = \begin{cases} 0 & \text{if \(\sim\) is numerical (or homological) equivalence,}\\ J(k) \otimes \mathbb{Q} & \text{if \(\sim\) is rational equivalence.}\end{cases} \] As a corollary one deduces that the category of abelian varieties of \(k\) up to isogeny is equivalent to the full subcategory of \({\mathcal M}_ k\) whose objects are direct summands of motives of the form \(h^ 1(X)\), where \(X\) is of dimension one. Another corollary, already mentioned before, says that \({\mathcal M}^{\text{rat}}_ k\) will not be abelian if \(k\) is not contained in the closure of a finite field. The proof of this fact depends on the existence of an elliptic curve over \(k\) with a \(k\)-rational point of infinite order.
The fourth section is concerned with the construction of Murre’s Picard and Albanese’s motives (for rational equivalence!) for a smooth projective \(d\)-dimensional variety \(X\) over \(k\). It consists in the explicit construction of projectors \(p_ 1\) and \(p_{2d-1} = {}^ t p_ 1\) such that \(p_ 0\), \(p_ 1\), \(p_{2d-1}\) and \(p_{2d}\) are orthogonal idempotents. One also obtains a Lefschetz type isomorphism and a description of the Chow groups of \(h^ 1(X)\) and \(h^{2d-1}(X)\). In particular, for a surface \(X\) one obtains a complete list of the Chow groups of the constituent parts \(h^ i(X)\), \(i = 0,1,2,3,4\), of the motive \(h(X)\) of \(X\). For \(h^ 2(X) = (X, p_ 2,0)\) one takes \(p_ 2 = \text{id} - p_ 0 - p_ 1 - p_ 3 - p_ 4\).
The fifth section deals with work of Deninger and Murre and of Künnemann on motives of abelian varieties (or, more generally, abelian schemes). For these varieties one has a canonical decomposition in \({\mathcal M}_ k = {\mathcal M}^{\text{rat}}_ k\) of the form \[ h(X) = \bigoplus_{i = 0}^{2g} h^ i(X)^{\text{can}}, \] where \(g = \dim X\), and where the \(h^ i (X)\) are the ‘\(n^ i\)-eigenspaces’ for multiplication by \(n\) on \(X\). One has for every \(i \geq 0\) isomorphisms \(\bigwedge^ i h^ 1(X)^{\text{can}}\overset \sim \longrightarrow h^ i(X)^{\text{can}}\). Also, for the class \(\xi\in A^ 1(X)\) of a symmetric line bundle on \(X\), one gets \(h(X) \overset {\xi^{2g - 1}} \longrightarrow h(X) \otimes \mathbb{L}^{1-g}\) and this fits in a commutative diagram with \(h^ i(X) \overset \sim \longrightarrow h^{2g-1} \otimes \mathbb{L}^{1-g}\), where the indexed \(h\)’s map into \(h(X)\). Finally, the canonical projectors defining the \(h^ i(X)^{\text{can}}\) coincide with the \(p_ i\)’s of the fourth section for \(i = 0,1,2g - 1,2g\) whenever these are constructed by means of a symmetric class \(\xi\) of a very ample line bundle on \(X\).
In the last sections several less classical topics are briefly discussed: (i) relative motives; (ii) the general picture englobing numerical, homological and rational equivalence and the respective categories of motives. This is mainly conjectural, depending on Grothendieck’s standard conjectures and conjectures due to Beilinson and Murre on a filtration of Chow groups; (iii) the formalism in terms of derived categories: there should exist a derived category of mixed motives containing \({\mathcal M}_ k^{\text{num}}\) as the abelian, semi-simple subcategory of ‘pure complexes’ of any weight concentrated in degree zero, and containing \({\mathcal M}^{\text{rat}}_ k\) as the subcategory of ‘pure complexes of weight zero’.
For the entire collection see [Zbl 0788.00053].

MSC:

14A20 Generalizations (algebraic spaces, stacks)
14C25 Algebraic cycles
14F99 (Co)homology theory in algebraic geometry
14C15 (Equivariant) Chow groups and rings; motives
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