aa r X i v : . [ m a t h . AG ] N ov Projectors on the intermediate algebraic Jacobians
Charles Vial
Abstract
Let X be a complex smooth projective variety of dimension d . Under some as-sumption on the cohomology of X , we construct mutually orthogonal idempotents in CH d ( X × X ) ⊗ Q whose action on algebraically trivial cycles coincides with the Abel-Jacobi map. Such a construction generalizes Murre’s construction of the Albanese andPicard idempotents and makes it possible to give new examples of varieties admittinga self-dual Chow-K¨unneth decomposition satisfying the motivic Lefschetz conjectureas well as new examples of varieties having a Kimura finite dimensional Chow motive.For instance, we prove that fourfolds with Chow group of zero-cycles supported on acurve (e.g. rationally connected fourfolds) have a self-dual Chow-K¨unneth decomposi-tion which satisfies the motivic Lefschetz conjecture and consequently Grothendieck’sstandard conjectures. We also prove that hypersurfaces of very low degree are Kimurafinite dimensional. Introduction
Let X be a smooth projective variety of dimension d over an algebraically closed field k ⊂ C . The Chow group CH i ( X ) of cycles of dimension i on X is the Q -vectorspacegenerated by i -cycles on X modulo rational equivalence. Given ∼ an equivalence relationon cycles, CH i ( X ) ∼ denotes those cycles which are ∼
0. In this paper, ∼ will either bealgebraic, homological or numerical equivalence. All three equivalence relations agree onzero-cycles and are spanned by the zero cycles of degree zero.Being able to exhibit cycles in CH d ( X × X ) with appropriate action on the homologyof X is essential to Grothendieck’s theory of pure motives. As discussed for instance in[11], being able to exhibit cycles in CH d ( X × X ) which are idempotents is a prerequisite tothe understanding of Chow groups as part of the framework of the Bloch-Beilinson-Murrephilosophy. Roughly speaking, such a framework predicts that the Chow groups of X should be controlled by the cohomology of X . In this paper we address a question of aslightly different nature as whether the Chow groups of X dictate its Chow motive. Ofcourse, we do not answer such a question in generality. However, we completely answer thisquestion in the case when the Chow groups of X are generated by the Chow groups of zero-cycles of curves. For this purpose we construct appropriate idempotents in CH d ( X × X ).In the spirit of the BBM philosophy, work of Esnault and Levine [6] (and Jannsen [11] in1he case of points) shows that if the Chow groups of X are generated by the Chow groupsof curves, then the cohomology of X is generated by the cohomology of curves. Here, asa consequence of the construction of appropriate idempotents, we show that if the Chowgroups of X are generated by the Chow groups of curves, then not only is the cohomologyof X generated by the cohomology of curves but the Chow motive of X is generated by theChow motives of curves (see theorem 4 below). In particular, this complements Esnaultand Levine’s theorem by showing that the Chow motive of X is finite dimensional in thesense of Kimura [14]. The basic properties of pure motives are exposed in [20] and the(covariant) notations we use are those of [13].Murre [17] constructed mutually orthogonal idempotents Π and Π d − in CH d ( X × X )called respectively the Albanese projector and the Picard projector. Such idempotentssatisfy the following properties. • Π = t Π d − . • (Π ) ∗ H ∗ ( X ) = H ( X ) and (Π d − ) ∗ H ∗ ( X ) = H d − ( X ). • (Π ) ∗ CH ∗ ( X ) = (Π ) ∗ CH ( X ) ≃ Alb X ( k ) ⊗ Q . • (Π d − ) ∗ CH ∗ ( X ) = CH d − ( X ) hom ≃ Pic X ( k ) ⊗ Q .Scholl [20] then showed that it is possible to modify slightly the construction of theseidempotents in order to, in addition, have a Lefschetz isomorphism : • The map ι ∗ ι ∗ : ( X, Π d − , → ( X, Π , d −
1) is an isomorphism of Chow motives.Here ι : C → X is a smooth linear section of dimension one of X .In this paper, we wish to generalize Murre’s construction in the following sense : we wishto construct mutually orthogonal idempotents Π i +1 ,i in CH d ( X × X ) which, in homology,define projectors on the largest subHodge structure of H i +1 ( X ) generated by the H ’s ofcurves. Here H k ( X ) := H k ( X ( C ) , Q ) which is isomorphic to H d − k ( X ( C ) , Q ). We offertwo different constructions.The first construction is exposed in the first section. It is defined for all smooth projec-tive varieties X but we cannot show that the idempotents constructed there act appropri-ately in homology without making some assumptions on X . In some sense the idempotentsconstructed there lift the largest submotive of a curve contained in the numerical motive of X . What is needed is Jannsen’s semi-simplicity theorem [10] in order to produce idempo-tents modulo numerical equivalence, and then a lifting property from numerical equivalenceup to rational equivalence (proposition 1.1).The second construction, which is much more precise, gives the required idempotentsbut depends on an assumption on the cohomology of X which we describe below. Let’s2efine N i H i ( X ) to be the image of the rational cycle class map cl i : CH i ( X ) → H i ( X )and N i H i +1 ( X ) := X Im (cid:0) Γ ∗ : H ( C ) → H i +1 ( X ) (cid:1) , where the sum runs through all smooth projective curves C and through all correspondencesΓ ∈ CH i +1 ( C × X ). The use of the notation N i H i +1 ( X ) is not arbitrary since it can beshown that this subgroup of H i +1 ( X ) is spanned by those classes that vanish in the opencomplement of some subvariety of X of dimension i + 1. The group N i H i +1 ( X ) is thusthe last step of the coniveau filtration on H i +1 ( X ).Given an integer i , the assumption we need on X in order to construct the idempotentthat we will denote Π i, ⌊ i/ ⌋ is that the cup product pairing H d − i ( X ) × H i ( X ) → Q restrictsto a non degenerate pairing on N ⌊ (2 d − i ) / ⌋ H d − i ( X ) × N ⌊ i/ ⌋ H i ( X ). We begin the secondsection by showing in lemma 2.1 that such pairings are non degenerate for a large class ofvarieties. Lemma 2.1 also shows that these pairings are expected to be non degenerate forall smooth projective varieties if one believes in Grothendieck’s standard conjectures.The construction of the projectors Π i,i is unsurprising and is usually used to extractthe N´eron-Severi group N S i ( X ) out of CH i ( X ) : Theorem 1.
Let i be an integer. Assume that the pairing N d − i H d − i ( X ) × N i H i ( X ) → Q is non degenerate. Then there exist idempotents Π i,i and Π d − i,d − i in CH d ( X × X ) such that • Π i,i = t Π d − i,d − i . • (Π i,i ) ∗ H ∗ ( X ) = N i H i ( X ) . • CH i ( X ) hom = Ker (cid:0) Π i,i : CH i ( X ) → CH i ( X ) (cid:1) . • The Chow motive ( X, Π i,i , is isomorphic to ( L ⊗ i ) ⊕ d i where d i = dim N i H i ( X ) . • If i ≥ d there is a Lefschetz isomorphism of Chow motives ( X, Π i,i , → ( X, Π d − i,d − i , i − d ) given by intersecting i − d times with a smooth hyperplane section of X . We now turn to the construction of the projectors Π i +1 ,i . In particular, our con-struction gives a motivic interpretation of the Abel-Jacobi map to the algebraic part ofthe intermediate Jacobians. Write J ai ( X ) for the image of the Abel-Jacobi map AJ i : CH i ( X ) Z alg → J i ( X ( C )), it is an algebraic torus defined over k . Theorem 2.
Let i be an integer. Assume that the pairing N d − i − H d − i − ( X ) × N i H i +1 ( X ) → Q is non degenerate. Then there exist idempotents Π i +1 ,i and Π d − i − ,d − i − in CH d ( X × X ) such that • Π i +1 ,i = t Π d − i − ,d − i − . • (Π i +1 ,i ) ∗ H ∗ ( X ) = N i H i +1 ( X ) . Ker (cid:0) AJ i : CH i ( X ) alg → J ai ( X )( k ) ⊗ Q (cid:1) = Ker (cid:0) Π i +1 ,i : CH i ( X ) alg → CH i ( X ) alg (cid:1) . • The Chow motive ( X, Π i +1 ,i , is isomorphic to h ( J ai ( X ))( i ) . • If i + 1 ≥ d there is a Lefschetz isomorphism of Chow motives ( X, Π i +1 ,i , → ( X, Π d − i − ,d − i − , i + 1 − d ) given by intersecting i + 1 − d times with a smoothhyperplane section of X . These generalize Murre’s construction of the Albanese and Picard projectors (Π , andΠ d − ,d − respectively) because in the cases i = 0 or i = d − N H ( X ) = H ( X )and N d − H d − ( X ) = H d − ( X ). The pairing N d − i − H d − i − ( X ) × N i H i +1 ( X ) → Q is thus just the cup product pairing between H d − ( X ) and H ( X ) and is always nondegenerate.Finally lemma 2.1 shows that the above pairings are all non degenerate for curves,surfaces, abelian varieties, complete intersections, uniruled threefolds, rationally connectedfourfolds and any smooth hyperplane section, product and finite quotient thereof. Forthose varieties X for which those idempotents can be constructed for all i we can show,thanks to the non-commutative Gram-Schmidt process of lemma 2.12, that it is possibleto choose the idempotents of theorems 1 and 2 to be pairwise orthogonal : Theorem 3.
If the pairings N ⌊ (2 d − i ) / ⌋ H d − i ( X ) × N ⌊ i/ ⌋ H i ( X ) → Q are non degeneratefor all i then the idempotents of theorems 1 and 2 can be chosen to be pairwise orthogonal. The second section is then devoted to the proof of these theorems.In the third section, we compute the Chow motive of those varieties whose Chowgroups are all representable. We say that CH i ( X ) alg is representable if there exists acurve C and a correspondence Γ ∈ CH i +1 ( C × X ) such that CH i ( X ) alg = Γ ∗ CH ( C ) alg .We show that if X is a variety whose Chow groups are all representable then the pairings N ⌊ (2 d − i ) / ⌋ H d − i ( X ) × N ⌊ i/ ⌋ H i ( X ) → Q are non degenerate for all i . We then use theorems1, 2 and 3 to compute the Chow motive of varieties having representable Chow groups.Informally, Esnault and Levine [6] showed that if the Chow groups of a variety X areall representable then the cohomology of X is generated by the cohomology of curves.Conversely Kimura [14] proved that if the cohomology groups of X are generated by thecohomology of curves and if the Chow motive of X is finite dimensional (See [14] fora definition) then the Chow groups of X are representable. Here we prove a strongerstatement : Theorem 4.
Let k ⊆ C be an algebraically closed field. Let X be a smooth projectivevariety of dimension d over k . Write X C := X × Spec k Spec C and b j := dim H j ( X ) . Thefollowing statements are equivalent.1. h ( X ) = ⊕ h (Alb X ) ⊕ L ⊕ b ⊕ h ( J a ( X ))(1) ⊕ ( L ⊗ ) ⊕ b ⊕· · ·⊕ h ( J ad − ( X ))( d − ⊕ L ⊗ d . . The cycle class maps cl i : CH i ( X ) → H i ( X ) and the rational Deligne cycle classmaps cl D i : CH i ( X C ) → H D i ( X, Q ( i )) are surjective for all i and h ( X ) is finitedimensional.3. The rational Deligne cycle class maps cl D i : CH i ( X C ) → H D i ( X, Q ( i )) are injectivefor all i .4. The Chow groups CH i ( X C ) alg are representable for all i . Esnault and Levine [6] proved that if the total Deligne cycle class map of X is injectivethen it is surjective. Theorem 4 gives thus a better insight to the link between injectivityand surjectivity of cycle class maps.The proof of the theorem goes as follows. The first statement is certainly the strongest,i.e. it implies the three others. The equivalence (3 ⇔
4) is certainly known but we couldn’tfind a reference for (4 ⇒ § ⇒ ⇒
1) which settles the theorem and which we give in § ⇒
1) in § i,i and Π i +1 ,i .As an immediate corollary we get a generalization of a result by Jannsen [11, Th. 3.6]who proved that if the total cycle class map of X is injective then it is surjective. This wasalso proved by Kimura [15]. Theorem 5.
Let k ⊆ C be an algebraically closed field. Let X be a smooth projectivevariety of dimension d over k . The following statements are equivalent.1. h ( X ) = L di =0 ( L ⊗ i ) ⊕ b i .2. The rational cycle class map cl : CH ∗ ( X C ) → H ∗ ( X ) is surjective and h ( X ) is finitedimensional.3. The rational cycle class maps cl : CH ∗ ( X C ) → H ∗ ( X ) is injective.4. The Chow groups CH i ( X C ) are finite dimensional Q -vector spaces for all i . Again this theorem makes more precise the link between injectivity and surjectivity ofcycle class maps.In the fourth and last section we are interested in using our construction of idempotentsto give new examples of varieties for which we can compute explicitly a Chow-K¨unnethdecomposition of the diagonal. Such examples include 3-folds X satisfying H ( X, Ω X ) = 0(e.g. Calabi-Yau 3-folds), rationally connected 4-folds, and 4-folds admitting a rationalmap to a curve with rationally connected general fiber. We also prove that such varietiessatisfy the motivic Lefschetz conjecture and hence Grothendieck’s standard conjectures.5e are also interested in giving new examples of varieties whose Chow motives are finitedimensional in the sense of Kimura [14]. These will be given by smooth hyperplane sectionsof hypersurfaces covered by a family of linear projective varieties of dimension ⌊ n/ ⌋ in P n +1 . These were considered by Esnault, Levine and Viehweg [7] and also subsequentlyby Otwinowska [18] and include hypersurfaces of very small degree, e.g. cubic 5-folds,5-folds which are the smooth intersection of a cubic and a quadric and 7-folds which arethe smooth intersection of two quadrics. Other examples are given by rationally connectedthreefolds, a case which was treated by Gorchinskiy and Guletskii in [8].Let’s mention that the construction given in the first section is used in [22] to provea generalization of the implication (4 ⇒
1) in theorem 4 to the case of Chow motiveswith representable Chow groups. The proof given there does not involve any cohomologytheory, except implicitly through the use of Jannsen’s semi-simplicity theorem whose proofrequires the existence of a “good” cohomology theory. In [21], we prove Murre’s conjecturesfor the varieties considered in §
4. We also refer to [21, §
2] for some statements that do notinvolve the cohomology (or the Chow groups) of X in all degrees. Aknowledgements.
Thanks to Burt Totaro for useful comments. This work is sup-ported by a Nevile Research Fellowship at Magdalene College, Cambridge and an EPSRCPostdoctoral Fellowship under grant EP/H028870/1. I would like to thank both institu-tions for their support.
The coniveau filtration on numerical motives.
Let k be any field and X a smoothprojective variety over k of dimension d . We refer to [13, § k is denoted M and the categoryof numerical motives over k is denoted ¯ M . The reduction modulo numerical equivalenceof a cycle γ ∈ CH k ( X ) is denoted ¯ γ . A fundamental result of Jannsen [10] states thatthe category of numerical motives is abelian semi-simple. In particular if f : N → M is amorphism of numerical motives with M = ( X, p, n ), Jannsen’s theorem gives the existenceof a correspondence π ∈ End k ( M ) such that Im f ≃ ( X, π, n ).It is thus possible to define a coniveau filtration on numerical motives as in [1, § § N j M := P (cid:0) f : ¯ h ( Y )( j ) → M (cid:1) where the sum runs through all smoothprojective varieties Y and all morphisms f ∈ Hom k (¯ h ( Y )( j ) , M ) and where ¯ h ( Y )( j ) denotesthe numerical motive of Y tensored j times by the Lefschetz motive.Let’s imagine for a moment that Grothendieck’s standard conjecture B (cf. [1, 5.2.4.1])is true. Then [1, 5.4.2.1] each numerical motive M has a weight decomposition that wewrite M = L i M i . Furthermore, for weight reasons N j M i = P (cid:0) f : ¯ h i − j ( Y )( j ) → M i (cid:1) .6nother consequence of Grothendieck’s standard conjecture B is that if i : Z → Y is asmooth hyperplane section of dimension i − j of Y then i ∗ : ¯ h i − j ( Z ) → ¯ h i − j ( Y ) issurjective [1, 5.2.5.1]. Therefore we have N j M i = P Im (cid:0) f : ¯ h i − j ( Y )( j ) → M i (cid:1) , where thesum runs through all smooth projective varieties Y with dim Y = i − j and through allmorphisms f ∈ Hom k (¯ h i − j ( Y )( j ) , M i ). The idempotents ¯ π j,j and ¯ π j +1 ,j . Let’s now forget about the standard conjectures.We know that points and curves have a weight decomposition [1, 4.3.2]; it is thereforenatural for any integer j and for any numerical motive M to consider the following directsummands of M : M j,j := X Im (cid:0) f : ¯ h (Spec k )( j ) → M (cid:1) and M j +1 ,j := X Im (cid:0) f : ¯ h ( C )( j ) → M (cid:1) where the first sum runs through all morphisms f ∈ Hom k (¯ h (Spec k ) , M ) and the secondsum runs through all curves C and through all morphisms f ∈ Hom k (¯ h ( C ) , M ). Thus inparticular there exist for all integers j corespondences π j,j and π j +1 ,j in CH d ( X × X )such that M j,j = ( X, ¯ π j,j ,
0) and M j +1 ,j = ( X, ¯ π j +1 ,j , A lifting property.
We denote by M (resp. M ) the full thick subcategory of M generated by the Chow motives of points (resp. the h ’s of smooth projective curves over k ). For a motive P ∈ M , let ¯ P denote its image in ¯ M . (This notation is abusive sincewe previously denoted numerical motives with a bar and it is not known if all numericalmotives admit a lift to rational equivalence). Let’s also write ¯ M (resp. ¯ M ) for theimage of M (resp. M ) in ¯ M . The functors M → ¯ M and M → ¯ M are equivalenceof categories (see [20, corollary 3.4]) and as such the categories M and M are abeliansemi-simple. Proposition 1.1.
Let M be an object in M (resp. in M ). Let N be any motive in M .Then any morphism f : M → N induces a splitting N = N ⊕ N with N isomorphic toan object in M (resp. in M ) and ¯ N ≃ Im ¯ f .Proof. The morphism M → N induces a morphism ¯ M → ¯ N and it is known that anymorphism in an abelian semi-simple category is a direct sum of a zero morphism and of anisomorphism (cf. [2, A.2.13]). Let’s thus write¯ M = ¯ M ⊕ ¯ M f ⊕ −→ ¯ N ⊕ ¯ N = ¯ N where ¯ f is an isomorphism ¯ M ∼ −→ ¯ N . The composition ( ¯ f − ⊕ ◦ ( ¯ f ⊕ ∈ End( ¯ M )is therefore equal to the projector ¯ M → ¯ M → ¯ M . Let then g : N → M be any lift of¯ f − ⊕ N → ¯ M and let M be any lift of ¯ M . Then g ◦ f ∈ End( M ). But it is a fact thatEnd( M ) = End( ¯ M ). Therefore, g ◦ f is a projector on M . We now claim that f ◦ g ◦ f ◦ g defines a projector in End( N ) onto an object isomorphic to M . Indeed,( f ◦ g ◦ f ◦ g ) ◦ ( f ◦ g ◦ f ◦ g ) = f ◦ (cid:0) ( g ◦ f ) ◦ ( g ◦ f ) ◦ ( g ◦ f ) (cid:1) ◦ g = f ◦ ( g ◦ f ) ◦ g = f ◦ g ◦ f ◦ g N g / / M f / / ! ! CCCCCCCC N g / / M f / / ! ! CCCCCCCC N g / / M f / / ! ! CCCCCCCC N g / / M f / / NM / / = = {{{{{{{{ M / / = = {{{{{{{{ M = = {{{{{{{{ showing that indeed f ◦ g ◦ f ◦ g projects onto M (since it has a retraction). The idempotents π j,j and π j +1 ,j . Proposition 1.1 shows that it is actually possibleto choose the correspondences π j,j and π j +1 ,j above to be idempotents in CH d ( X × X ). Inother words, proposition 1.1 shows that it is possible to define direct summands ( X, π j,j , X, π j +1 ,j ,
0) of the Chow motive h ( X ) of X whose reduction modulo numericalequivalence are the direct summands M j,j and M j +1 ,j defined above.We won’t be giving the details here but it can be shown that, if Grothendieck’s standardconjecture B is true for all smooth projective varieties, then the idempotents π j,j and π j +1 ,j constructed here coincide modulo homological equivalence with the idempotentsΠ j,j and Π j +1 ,j of § A remark about the K¨unneth projectors.
We would like to explain how it is possibleto construct cycles whose numerical classes are the expected K¨unneth projectors, i.e. whosehomological classes are expected to be the projectors H ∗ ( X ) → H i ( X ) → H ∗ ( X ). Weproceed by induction on d = dim X . If X = Spec k , we define π X to be the cycle X × X inside X × X . Suppose we have constructed projectors modulo numerical equivalence π Y , π Y , . . . , π Y Y ) − for all smooth projective varieties Y of dimension dim Y < d . Then,for all i ∈ { , . . . , d − } , we define the cycle π Xi ∈ CH d ( X × X ) / num to be the projectorsuch that ( X, π Xi ,
0) = [ f : Y → X Im (cid:0) f ∗ : ( Y, π Yi , → ¯ h ( X ) (cid:1) , where the sum runs through all smooth projective varieties Y of dimension i and all mor-phisms f : Y → X . We then set π X d − i = t π Xi and π Xd = id X − P i = d π Xi .If Grothendieck’s standard conjecture B is true, then it can be checked that those definethe expected K¨unneth projectors. Π i,i and Π i +1 ,i In this section, we fix an algebraically closed field k with an embedding k ֒ → C and weprove theorems 1, 2 and 3. We start with a lemma which shows that many varieties dosatisfy the assumptions of these theorems. 8et X be a d -dimensional smooth projective variety over k . Let ι : H → X be asmooth hyperplane section of X and let Γ ι ∈ CH d − ( H × X ) be its graph and let t Γ ι be the transpose of Γ ι . We define L := Γ ι ◦ t Γ ι ∈ CH d − ( X × X ). The hard Lefschetztheorem asserts that the map L i : H d + i ( X ) → H d − i ( X ) given by intersecting i times with H is an isomorphism for all i ≥
0. The variety X is said to satisfy property B if theinverse morphism is induced by an algebraic correrspondence for all i ≥
0. It is one ofGrothendieck’s standard conjectures that all smooth projective varieties should satisfy B.
Lemma 2.1.
Let i be an integer in { d, . . . , d } . The cup product pairing N ⌊ (2 d − i ) / ⌋ H d − i ( X ) × N ⌊ i/ ⌋ H i ( X ) → Q is non degenerate in either of the following cases: • X satisfies property B. • N ⌊ i/ ⌋ H i ( X ) = H i ( X ) .In particular the pairing N d − H d − ( X ) × N H ( X ) → Q is non degenerate for all X .Proof. In the case X satisfies property B, the Hodge index theorem is a crucial ingredientand the lemma is a special case of [21, Prop. 1.4]. The other case is obvious.Therefore the results in this section hold for curves, surfaces, abelian varieties, completeintersections, uniruled threefolds, rationally connected fourfolds and any smooth hypersur-face section, product or finite quotient thereof. All those varieties satisfy property B.Lemma 3.1 shows that the results in this section also hold for varieties for which somecycle class map is surjective. We are given a smooth projective variety X over k of dimension d . By definition N i H i ( X )coincides with the image of the cycle class map CH i ( X ) → H i ( X ). For each integer i ,let d i = dim Q N i H i ( X ) and let P i be the disjoint union of d i copies of Spec k . Noticethat d i > N i H i ( X ) always contains the ( d − i )-fold intersection of a hyperplanesection. We then fix Γ i ∈ CH i ( P i × X ) such that(Γ i ) ∗ : H ( P i ) ≃ −→ N i H i ( X )is bijective. This amounts to fixing a basis of N i H i ( X ) = Im ( cl i : CH i ( X ) → H i ( X )).For each integer i , we also fix a smooth projective curve (not necessarily connected) C i and a correspondence Γ i +1 ∈ CH i +1 ( C i × X ) such that(Γ i +1 ) ∗ H ( C i ) = N i H i +1 ( X ) . C i,l be the connected components of C i and for all l let z i,l be a rational point on C i,l .Up to composing Γ i +1 with the correspondence ∆ C i − P l (cid:0) { z i,l } × C i,l + C i,l × { z i,l } (cid:1) ∈ CH ( C i × C i ), we can and we will assume moreover that(Γ i +1 ) ∗ H ( C i ) = (Γ i +1 ) ∗ H ( C i ) = 0 . In order to establish the Lefschetz isomorphism of theorems 1, 2 and 3 we will makeuse of the following easy lemma.
Lemma 2.2.
Let i be an integer in { d + 1 , . . . , d } and assume that the cup productpairing N ⌊ (2 d − i ) / ⌋ H d − i ( X ) × N ⌊ i/ ⌋ H i ( X ) → Q is non degenerate. Then L i − d : H i ( X ) → H d − i ( X ) maps isomorphically N ⌊ i/ ⌋ H i ( X ) to N ⌊ (2 d − i ) / ⌋ H d − i ( X ) .Proof. The non degeneracy assumption says in particular that the two Q vectorspaces N ⌊ i/ ⌋ H i ( X ) and N ⌊ (2 d − i ) / ⌋ H d − i ( X ) have same dimension. The Lefschetz isomorphism L restricts to an injective map N ⌊ i/ ⌋ H i ( X ) → H d − i ( X ) and, by definition of N , maps N ⌊ i/ ⌋ H i ( X ) into N ⌊ (2 d − i ) / ⌋ H d − i ( X ). Remark 2.3.
In fact if the pairing N ⌊ (2 d − i ) / ⌋ H d − i ( X ) × N ⌊ i/ ⌋ H i ( X ) → Q is non degen-erate, then more is true. Namely, as a consequence of the Lefschetz isomorphisms of propo-sitions 2.4 and 2.8, we have that the isomorphism L i − d : N ⌊ i/ ⌋ H i ( X ) → N ⌊ (2 d − i ) / ⌋ H d − i ( X )has its inverse induced by a correspondence.If for an integer i such that 2 i ∈ { d, . . . , d } , the cup product pairing N d − i H d − i ( X ) × N i H i ( X ) → Q is non degenerate, lemma 2.2 makes it possible to furthermore assumethat P i = P d − i and Γ d − i = L i − d ◦ Γ i .Likewise if for an integer i such that 2 i + 1 ∈ { d, . . . , d } , the cup product pairing N d − i − H d − i − ( X ) × N i H i +1 ( X ) → Q is non degenerate, lemma 2.2 makes it possibleto furthermore assume that C i = C d − i − and Γ d − i − = L i +1 − d ◦ Γ i +1 . In this paragraph we consider an integer i with 2 i ≥ d and a smooth projective variety X of dimension d for which the pairing N d − i H d − i ( X ) × N i H i ( X ) → Q is non degenerate.The correspondence Γ d − i := L i − d ◦ Γ i induces by duality a bijective map( t Γ d − i ) ∗ : (cid:0) N d − i H d − i ( X ) (cid:1) ∨ ≃ −→ H ( P i ) ∨ . By the non degeneracy assumption (cid:0) N d − i H d − i ( X ) (cid:1) ∨ identifies with N i H i ( X ) and H ( P i ) ∨ identifies with H ( P i ). Therefore the composition t Γ d − i ◦ Γ i induces a Q -linear iso-morphism H ( P i ) ≃ −→ H ( P i ). It is then clear that there exists a correspondence γ i ∈ H ( P i × P i ) such that γ i ◦ t Γ d − i ◦ Γ i acts as identity on H ( P i ). Because L = t L wealso have γ i = t γ i . We then set Π i,i := Γ i ◦ γ i ◦ t Γ d − i ∈ CH d ( X × X ) . Since γ i ◦ t Γ d − i ◦ Γ i = id ∈ CH ( P i × P i ), it is clear that Π i,i is an idempotent andthat it induces the projector H ∗ ( X ) → N i H i ( X ) → H ∗ ( X ) in homology. Also, it is clearthat Π d − i,d − i := t Π i,i (Notice that if 2 i = d then Π d,d/ = t Π d,d/ ) defines an idempotentwhich induces the projector H ∗ ( X ) → N d − i H d − i ( X ) → H ∗ ( X ) in homology. Proposition 2.4.
The correspondence Π d − i,d − i ◦ L i − d ◦ Π i,i : ( X, Π i,i , → ( X, Π d − i,d − i , i − d ) is an isomorphism of Chow motives.Proof. Using the identities L = t L , γ i = t γ i , Π d − i,d − i = t Π i,i and the fact thatΠ i,i = Γ i ◦ γ i ◦ t Γ i ◦ t L i − d is an idempotent, one can easily check that Π i,i ◦ Γ i ◦ γ i ◦ t Γ i ◦ Π d − i,d − i is the inverse of Π d − i,d − i ◦ L i − d ◦ Π i,i . Proposition 2.5.
Let Π i ∈ CH d ( X × X ) be an idempotent which factors through a zero-dimensional variety P i as Π i = Γ ◦ α with Γ ∈ CH i ( P i × X ) and α ∈ CH d − i ( X × P i ) , andwhose action on H ∗ ( X ) is the orthogonal projection on N i H i ( X ) . Then the Chow motive ( X, Π i , is isomorphic to ( L ⊗ i ) ⊕ d i .Proof. The cycle class map CH ( P i ) → H ( P i ) is an isomorphism. Let π := α ◦ Γ ∈ CH ( P i × P i ). By functoriality of the cycle class map, we see that π is an idempotentsuch that ( P i , π,
0) = ⊕ d i . The correspondence Γ is an element of CH i ( P i × X ) =Hom k (cid:0) h ( P i )( i ) , h ( X ) (cid:1) and it can easily be checked that the correspondence Π i ◦ Γ ◦ π ∈ Hom k (cid:0) ( P i , π, i ) , ( X, Π i , (cid:1) is an isomorphism with inverse π ◦ α ◦ Π i . Proposition 2.6.
Let Q i be a correspondence in CH d ( X × X ) such that Q i acts as theidentity on N i H i ( X ) and such that Q i is supported on X × Z with Z a subvariety of X of dimension i . Then CH i ( X ) hom = Ker (cid:0) Q i : CH i ( X ) → CH i ( X ) (cid:1) . In particular CH i ( X ) hom = Ker (cid:0) Π i,i : CH i ( X ) → CH i ( X ) (cid:1) . Proof.
By functoriality of the cycle class map, we have a commutative diagram CH i ( X ) cl i (cid:15) (cid:15) / / CH ( Z ) ≃ (cid:15) (cid:15) / / CH i ( X ) cl i (cid:15) (cid:15) H i ( X ) / / H ( Z ) / / H i ( X )The composition of the two arrows of the top row is the map induced by Q i and thecomposition of the two arrows of the bottom row is the identity on Im ( cl i ). The propositionfollows easily. 11 .3 Proof of theorem 2 The Albanese and the Picard varieties.
Let X be a smooth projective variety overa field k . The Albanese variety attached to X and denoted Alb X is an abelian varietyuniversal for maps X → A from X to abelian varieties A sending a fixed point x ∈ X to 0 ∈ A . The Picard variety Pic X of X is the abelian variety parametrizing numericallytrivial line bundles on X (i.e. those with vanishing Chern class). These define respectivelya covariant and a contravariant functor from the category of smooth projective varieties tothe category of abelian varieties.The abelian varieties Alb X and Pic X are dual and are isogenous in the following way.Let C be a curve which is a smooth linear section of X . Then the mapΨ : Pic X → Pic C Θ −→ Alb C → Alb X is an isogeny, where Θ is the map induced by the theta-divisor on the curve C .The following proposition is essential to the construction of the idempotents Π i +1 ,i . Proposition 2.7 ( cf. th. 3.9 and prop. 3.10 of [20]) . Let Y and Z be connected smoothprojective varieties and let ζ ∈ CH ( Y ) and η ∈ CH ( Z ) be -cycles of positive degree.Then, there is an isomorphism Ω : Hom(Alb Y , Pic Z ) ⊗ Q ≃ −→ { c ∈ CH ( Y × Z ) / c ( ζ ) = t c ( η ) = 0 } . Moreover, Ω is functorial in the following sense. Let φ : Y ′ → Y and ψ : Z ′ → Z bemorphisms of varieties and let ζ ′ and η ′ be positive -cycles on Y ′ and Z ′ with direct image ζ and η on Y and Z . If β : Alb Y → Pic Z is a homomorphism, then Ω(Pic ψ ◦ β ) = ψ ∗ ◦ Ω( β ) and Ω( β ◦ Alb φ ) = Ω( β ) ◦ φ ∗ , where Ω is taken with respect to the chosen -cycles. Intermediate Jacobians.
Given a smooth projective complex variety X , the i th inter-mediate Jacobian attached to X is the compact complex torus J i ( X ) = H i +1 ( X, C ) F i H i +1 ( X, C ) + H i +1 ( X, Z ) . It comes with a map AJ i : CH Z i ( X ) hom → J i ( X )defined on the integral Chow group called the i th Abel-Jacobi map which was thoroughlystudied by Griffiths [9]. In the cases i = 0 and i = dim X −
1, we recover the notionsof Albanese variety and Picard variety respectively. These intermediate Jacobians arehowever fairly different since they are of transcendental nature. While the Albanese and the12icard variety are algebraic tori, this is not the case in general for intermediate Jacobians.Precisely, let J alg i denote the maximal subtorus inside J i ( X ) whose tangent space is includedin H i +1 ,i ( X, C ). It is then a fact that J alg i is an abelian variety and that J alg i ( X ) = N iH H i +1 ( X, C ) N iH H i +1 ( X, C ) ∩ (cid:0) F i H i +1 ( X, C ) + H i +1 ( X, Z ) (cid:1) where N iH H i +1 ( X ) is the maximal sub-Hodge structure of H i +1 ( X ) contained in H i +1 ,i ( X, C ) ⊕ H i,i +1 ( X, C ). In particular, the intermediate Jacobian is algebraic if and only if H i +1 ( X, C )is concentrated in degrees ( i, i + 1) and ( i + 1 , i ). As a consequence of the horizon-tality of normal functions associated to algebraic cycles [9], the cycles in CH Z i ( X ) hom that are algebraically trivial map into J alg i ( X ) under the Abel-Jacobi map. The map CH Z i ( X ) alg → J alg i ( X ) is surjective if N i H i +1 ( X ) ⊇ N iH H i +1 ( X ) (the reverse inclusionalways holds), in particular if N i H i +1 ( X ) = H i +1 ( X ). In any case, let’s write J ai ( X )for the image of the map AJ i : CH Z i ( X ) alg → J alg i ( X ). It is an abelian subvariety of theabelian variety J alg i ( X ) which is defined the same way as J alg i ( X ) with N H replaced with N . We sum this up in the commutative diagram CH Z i ( X ) hom AJ i / / J i ( X ) CH Z i ( X ) alg AJ i / / / / ?(cid:31) O O J ai ( X ) (cid:31) (cid:127) / / J alg i ( X ) . ?(cid:31) O O Finally if X is defined over an algebraically closed subfield k of C , the image of thecomposite map CH Z i ( X ) alg → CH Z i ( X C ) alg → J alg i ( X C )defines an abelian variety over k that we denote J ai ( X ). The projectors Π i +1 ,i and Π d − i − ,d − i − . Given any abelian varieties A and B ,Hom( A, B ) denotes the group of homomorphisms from A to B . Recall that the category ofabelian varieties up to isogeny is the category whose objects are the abelian varieties andwhose morphisms are given by Hom( A, B ) ⊗ Z Q for any abelian varieties A and B . Thiscategory is abelian semi-simple, cf. [23].In the rest of this paragraph we consider an integer i with 2 i +1 ≥ d and a smooth projec-tive variety X of dimension d for which the pairing N d − i − H d − i − ( X ) × N i H i +1 ( X ) → Q is non degenerate. In particular the dual of J ad − i − ( X ) identifies with J ai ( X ). Lemma 2.2implies that the correspondence L i +1 − d induces an isogeny Λ : J ai ( X ) → J ad − i − ( X ) andbecause L = t L we have Λ = Λ ∨ , i.e. Λ is equal to its dual.13aking up what was said in § C i over k andcorrespondences Γ i +1 ∈ CH i +1 ( C i × X ) and Γ d − i − := L i +1 − d ◦ Γ i +1 ∈ CH d − i ( C i × X )such that both maps(Γ i +1 ) ∗ : H ( C i ) → N i H i +1 ( X ) and (Γ d − i − ) ∗ : H ( C i ) → N d − i − H d − i − ( X )are surjective and such that both maps act trivially on H ( C i ) and on H ( C i ). The cor-respondence Γ i +1 induces by functoriality of the Abel-Jacobi map a surjective homomor-phism (Γ i +1 ) ∗ : Alb C i ։ J ai ( X )as well as a homomorphism with finite kernel( t Γ i +1 ) ∗ ◦ Λ : J ai ( X ) ֒ → Pic C i . By semisimplicity of the category of abelian varieties up to isogeny, there exists α ∈ Hom( J ai ( X ) , Alb C i ) ⊗ Q such that (Γ i +1 ) ∗ ◦ α = id J ai ( X ) . Let’s considerΦ := α ◦ Λ − ◦ α ∨ ∈ Hom(Pic C i , Alb C i ) ⊗ Q so that ( ⋆ ) (Γ i +1 ) ∗ ◦ Φ ◦ ( t Γ i +1 ) ∗ ◦ Λ = id J ai ( X ) . We would now like to use proposition 2.7 in order to give an algebraic origin to Φ. De-composing C i into the disjoint union of its connected components C i,l , proposition 2.7 givesa functorial isomorphism between Hom(Alb C i , Pic C i ) ⊗ Q and { c ∈ CH ( C i × C i ) / c ( z i,l ) = t c ( z i,l ) = 0 } for z i,l the rational point on C i,l considered in § C i , Alb C i ) ⊗ Q which is not quite the Hom group in the statement of proposition 2.7.To correct this, let Θ denote the theta-divisor of the curve C i . Then under the isomorphismof proposition 2.7, Φ corresponds to a correspondence γ i +1 := Θ ◦ Γ ′ ◦ Θ − ∈ CH ( C i × C i )satisfying ( γ i +1 ) ∗ z i,l = ( t γ i +1 ) ∗ z i,l = 0 for all l . Because Φ = Φ ∨ we have γ i +1 = t γ i +1 . We now setΠ i +1 ,i := Γ i +1 ◦ γ i +1 ◦ t Γ d − i − = Γ i +1 ◦ γ i +1 ◦ t Γ i +1 ◦ L i +1 − d ∈ CH d ( X × X ) . By ( ⋆ ), the action of Π i +1 ,i on J ai ( X ) is given by id J ai ( X ) . The fact that Π i +1 ,i defines aprojector goes as follows. It is enough to prove that γ i +1 ◦ t Γ i +1 ◦ Λ ◦ Γ i +1 ◦ γ i +1 = γ i +1 . Thanks to proposition 2.7, it is actually enough to prove Φ ◦ ( t Γ i +1 ) ∗ ◦ L i +1 − d ∗ ◦ (Γ i +1 ) ∗ ◦ Φ = Φ : Pic C i → Alb C i . This last statement follows directly from ( ⋆ ).14ow because Π i +1 ,i acts as the identity on J ai ( X ) and because Γ i +1 and Γ d − i − acttrivially on homology classes of degree = 1, we see that the homology class of Π i +1 ,i is theprojector H ∗ ( X ) → N i H i +1 ( X ) → H ∗ ( X ).Finally we set Π d − i − ,d − i − := t Π i +1 ,i , which is licit in the case 2 d − i − d since in this case γ i +1 = t γ i +1 implies Π i +1 ,i = t Π i +1 ,i . It is then straightfor-ward that Π d − i − ,d − i − defines and idempotent that induces the projector H ∗ ( X ) → N d − i − H d − i − ( X ) → H ∗ ( X ). Proposition 2.8.
The correspondence Π d − i − ,d − i − ◦ L i +1 − d ◦ Π i +1 ,i : ( X, Π i +1 ,i , → ( X, Π d − i − ,d − i − , i + 1 − d ) is an isomorphism of Chow motives.Proof. Using the identities L = t L , γ i +1 = t γ i +1 , Π d − i − ,d − i − = t Π i +1 ,i and the factthat Π i +1 ,i = Γ i +1 ◦ γ i +1 ◦ t Γ i +1 ◦ t L i +1 − d is an idempotent, one can easily check thatΠ i +1 ,i ◦ Γ i +1 ◦ γ i +1 ◦ t Γ i +1 ◦ Π d − i − ,d − i − is the inverse of Π d − i − ,d − i − ◦ L i +1 − d ◦ Π i +1 ,i . Proposition 2.9.
Let Π i +1 ∈ CH d ( X × X ) be an idempotent which factors through acurve C i as Π i +1 = Γ ◦ α with Γ ∈ CH i +1 ( C i × X ) and α ∈ CH d − i ( X × C i ) , and whoseaction on H ∗ ( X ) is the orthogonal projection on N i H i +1 ( X ) . Then the Chow motive ( X, Π i +1 ,i , is isomorphic to h ( J ai ( X ))( i ) .Proof. The assumption on the homology class of Π i +1 implies that Π i +1 acts as theidentity on J ai ( X ) and acts as zero on H i ( X ). Therefore, by functoriality of the cycleclass map, α ◦ Γ ∈ CH ( C i × C i ) acts as zero on some positive degree zero-cycle ζ on C i .Now a consequence of proposition 2.7 is that given two abelian varieties J and J ′ over k ,there is a canonical identificationHom( J, J ′ ) ⊗ Q = Hom k (cid:0) h ( J ) , h ( J ′ ) (cid:1) . Because Π i +1 acts as the identity on J ai ( X ), α ◦ Γ defines an idempotent π ∈ End( h ( C i ))such that ( C i , π, ≃ h ( J ai ( X )).The correspondence Γ seen as a morphism of motives belongs to Hom k (cid:0) h ( C i )( i ) , h ( X ) (cid:1) .Let’s show that Π i +1 ,i ◦ Γ ◦ π ∈ Hom k (cid:0) ( C i , π, i ) , ( X, Π i +1 ,i , (cid:1) is an isomorphism. In fact, let’s show that its inverse is given by π ◦ α ◦ Π i +1 ,i , i.e that(Π ◦ Γ ◦ π ) ◦ ( π ◦ α ◦ Π) = Π and ( π ◦ α ◦ Π) ◦ (Π ◦ Γ ◦ π ) = π as correspondences,where for convenience we have dropped the subscripts “2 i + 1”. But then, this is obviousbecause Π = Γ ◦ α and π = α ◦ Γ are idempotents.
Proposition 2.10.
Let Q i +1 be a correspondence in CH d ( X × X ) such that Q i +1 acts asthe identity on N i H i +1 ( X ) and such that Q i +1 is supported on X × Z with Z a subvariety f X of dimension i + 1 . Then Ker (cid:0) AJ i : CH i ( X ) alg → J i ( X ) ⊗ Q (cid:1) = Ker (cid:0) Q i +1 : CH i ( X ) alg → CH i ( X ) alg (cid:1) . In particular
Ker (cid:0) AJ i : CH i ( X ) alg → J i ( X ) ⊗ Q (cid:1) = Ker (cid:0) Π i +1 , : CH i ( X ) alg → CH i ( X ) alg (cid:1) . Proof.
The assumptions on Q i +1 imply that the action of Q i +1 on CH i ( X ) factorsthrough CH ( e Z ) for some desingularization e Z → Z ; they also imply that the inducedaction of Q i +1 on J ai ( X ) is the identity.We have thus the commutative diagram CH i ( X ) alg AJ i (cid:15) (cid:15) A / / CH ( e Z ) alg ≃ (cid:15) (cid:15) B / / CH i ( X ) alg AJ i (cid:15) (cid:15) J ai ( X ) ⊗ Q / / Pic e Z ⊗ Q / / J ai ( X ) ⊗ Q where A and B are correspondences such that B ◦ A = Q i +1 . The inclusion Ker AJ i ⊆ Ker Π i +1 ,i follows from the commutativity of the diagram, which itself is a consequenceof the functoriality of the Abel-Jacobi map with respect to the action of correspondences.The reverse inclusion Ker Π i +1 ,i ⊆ Ker AJ i follows from the fact that the composite of thetwo lower horizontal arrows is the identity on Im AJ i = J ai ( X ) ⊗ Q . Remark 2.11.
An interesting question is to decide whether or not the action of anidempotent on homology determines its action on Chow groups. For example, givenidempotents π i,i and π i +1 ,i ∈ CH d ( X × X ) such that ( π i,i ) ∗ H ∗ ( X ) = N i H i ( X ) and( π i +1 ,i ) ∗ H ∗ ( X ) = N i H i +1 ( X ), do we have CH i ( X ) hom = Ker (cid:0) π i,i : CH i ( X ) → CH i ( X ) (cid:1) and Ker (cid:0) AJ i : CH i ( X ) alg → J ai ( X ) ⊗ Q (cid:1) = Ker (cid:0) π i +1 ,i : CH i ( X ) alg → CH i ( X ) alg (cid:1) ?It is shown in [21] that this is the case if X is finite dimensional in the sense of Kimura. In this section we are given a smooth projective variety X of dimension d for which thepairings are all non degenerate. As such, by theorems 1 and 2 we can define all theidempotents Π i,i and Π i +1 ,i . However these are not all necessarily pairwise orthogonal.We start with the following linear algebra lemma which makes it possible to modify theidempotents so as to make them pairwise orthogonal. Lemma 2.12.
Let V be a Q -algebra and let k be a positive integer. Let π , . . . , π n be idem-potents in V such that π j ◦ π i = 0 whenever j − i < k and j = i . Then the endomorphisms p i := (1 − π n ) ◦ · · · ◦ (1 − π i +1 ) ◦ π i ◦ (1 − π i − ) ◦ · · · ◦ (1 − π ) define idempotents such that p j ◦ p i = 0 whenever j − i < k + 1 and j = i . roof. Let j and i be such that j − i < k + 1 and look atΠ := π j ◦ (1 − π j − ) ◦ · · · ◦ (1 − π ) ◦ (1 − π n ) ◦ · · · ◦ (1 − π i +1 ) ◦ π i . Suppose first j < i . Because we have π r ◦ π s = 0 for all r < s , we immediately see thatΠ = 0. Suppose j = i , it is also easy to see that in this case Π = π i . Finally, suppose that i < j < i + k + 1. Because π r ◦ π s = 0 for all r < s + k , we can see after expanding Π thatΠ = π j ◦ π i − π j ◦ π i ◦ π i − π j ◦ π j ◦ π i = 0.In our case of concern, we get Theorem 2.13.
Let X be a smooth projective variety of dimension d . Let i < d be aninteger and let π , . . . , π i ∈ CH d ( X × X ) be idempotents such that ( π j ) ∗ H ∗ ( X ) = H j ( X ) for all ≤ j ≤ i . Let π d − j := t π j for ≤ j ≤ i . If π r ◦ π s = 0 for all ≤ r < s ≤ d ,then the non-commutative Gram-Schmidt process of lemma 2.12 gives mutually orthogonalidempotents { p j } j ∈{ ,...,i, d − i,... d } such that ( p j ) ∗ H ∗ ( X ) = H j ( X ) and p d − j := t p j for all j ∈ { , . . . , i, d − i, . . . d } . Moreover, we have isomorphisms of Chow motives ( X, π j ) ≃ ( X, p j ) for all j .Proof. In order to get mutually orthogonal idempotents, it is enough to apply lemma 2.122 i + 2 times. In order to prove the theorem, it suffices to prove each statement after eachapplication of the process of lemma 2.12. Everything is then clear, except perhaps for thelast statement. The isomorphism is simply given by the correspondence p j ◦ π j and itsinverse is π j ◦ p j . Proposition 2.14.
The projectors of theorems 1 and 2 satisfy • Π i,i ◦ Π j,j = 0 for i = j . • Π i +1 ,i ◦ Π j +1 ,j = 0 for | i − j | > . • Π i +1 ,i ◦ Π j,j = 0 for | i − j | > . • Π i,i ◦ Π j +1 ,j = 0 for | i − j | > .Proof. Look at the dimension of t Γ d − i ◦ Γ j . Proposition 2.15.
The projectors of theorems 1 and 2 satisfy • Π i − ,i − ◦ Π i +1 ,i = 0 for all i . • Π i,i ◦ Π i +1 ,i = 0 and Π i +1 ,i ◦ Π i +2 ,i +1 = 0 for all i . roof. For the first point we have t Γ d − i +1 ◦ Γ i +1 ◦ γ i +1 ∈ CH ( C i × C i − ) and thus thereexist rational numbers a l,l ′ such that t Γ d − i +1 ◦ Γ i +1 ◦ γ i +1 = P l,l ′ a l,l ′ [ C i,l × C i − ,l ′ ].This yields ( t Γ d − i +1 ◦ Γ i +1 ◦ γ i +1 ) ∗ z i,l = P l ′ a l,l ′ [ C i − ,l ′ ]. By definition of γ i +1 we alsohave ( γ i +1 ) ∗ z i,l = 0 for all l . Hence a l,l ′ = 0 for all l and all l ′ . Therefore t Γ d − i +1 ◦ Γ i +1 ◦ γ i +1 = 0.For the second point, up to transposing it is enough to prove one of the two equalities.Let’s prove the second one. We have t Γ d − i − ◦ Γ i +2 ∈ CH ( P i +1 × C i ). But then, because t γ i +1 acts trivially on z i.l ∈ CH ( C i,l ) for all l , we see that γ i +1 acts trivially on CH ( C i ).Therefore γ i +1 ◦ t Γ d − i − ◦ Γ i +2 = 0. Remark 2.16.
We have shown through the two previous propositions that t Γ d − j ◦ Γ i ◦ γ i =0 for j − i < j, ⌊ j/ ⌋ ◦ Π i, ⌊ i/ ⌋ = 0 for j − i < Remark 2.17.
The missing orthogonal relations are Π i +1 ,i ◦ Π i,i = 0, Π i +2 ,i +1 ◦ Π i +1 ,i =0 or Π i +1 ,i ◦ Π i − ,i − = 0. There is no reason that these should hold true for the idem-potents constructed in §§ Lemma 2.18. If i ≥ d then t Π j ◦ L i − d ◦ Π j ′ = 0 for j, j ′ ≥ i except in the case i = j = j ′ .Proof. Up to transposing we only have to prove t Π j ◦ L i − d ◦ Π j ′ = 0 for j ′ ≥ j ≥ i not allequal. In fact it is enough to prove γ j ◦ t Γ j ◦ L i − d ◦ Γ j ′ ◦ γ j ′ = 0 for j ′ ≥ j ≥ i not all equal.The correspondence γ j ◦ t Γ j ◦ L i − d ◦ Γ j ′ ◦ γ j ′ is a cycle of dimension j + j ′ − i in the Chowgroup of P ⌊ j ′ ⌋ × P ⌊ j ⌋ , C ⌊ j ′ ⌋ × P ⌊ j ⌋ , P ⌊ j ′ ⌋ × C ⌊ j ⌋ or C ⌊ j ′ ⌋ × C ⌊ j ⌋ depending on the parity of j and j ′ . Notice that j + j ′ − i ≥
1, and that j + j ′ − i = 1 implies that j ′ = i + 1 and j = i , and that j + j ′ − i = 2 implies that j and j ′ have same parity (in fact j = j ′ = i + 1or j ′ = j + 2 = i + 2). The proof that γ j ◦ t Γ j ◦ L i − d ◦ Γ j ′ ◦ γ j ′ = 0 in each of these cases isthen similar to the cases treated in the proof of the previous proposition. Proof of theorem 3.
We proceed by induction on k ≥ P k : There exist idempotents Π i ∈ CH d ( X × X ) for 0 ≤ i ≤ d such that • Π j ◦ Π i = 0 if j − i < k and j = i . • Π i satisfies the properties listed in theorem 1 for all i . • Π i +1 satisfies the properties listed in theorem 2 for all i . • The Π i ’s satisfy the conclusion of lemma 2.18.Clearly if property P d +1 holds, then the idempotents Π i are mutually orthogonal.(Actually it is enough to settle P by remark 2.17). If we set Π i := Π i, ⌊ i/ ⌋ , we see thanksto theorems 1 and 2, remark 2.16 and lemma 2.18 that property P holds. Let’s supposethat property P k holds and let’s prove that P k +1 holds.18e set P i := (1 −
12 Π d ) ◦ (1 −
12 Π d − ) ◦ · · · ◦ (1 −
12 Π i +1 ) ◦ Π i ◦ (1 −
12 Π i − ) ◦ · · · ◦ (1 −
12 Π ) . By lemma 2.12, these define idempotents such that P j ◦ P i = 0 if j − i < k + 1 and j = i ..It remains to check that P i enjoys the same properties as Π i .It is straightforward from the formula that we have t P i = P d − i . It is also straightfor-ward that P i induces the projector H ∗ ( X ) → N ⌊ i/ ⌋ H i ( X ) → H ∗ ( X ) in homology.Let’s now consider an integer i ≥ d and prove that the Lefschetz correspondence L i − d induces an isomorphism of Chow motives ( X, P i , → ( X, P d − i , i − d ). In fact, we aregoing to show that t P i ◦ L i − d ◦ P i admits P i ◦ Γ i ◦ γ i ◦ t Γ i ◦ t P i as an inverse, i.e. that( P i ◦ Γ i ◦ γ i ◦ t Γ i ◦ t P i ) ◦ ( t P i ◦ L i − d ◦ P i ) = P i and ( t P i ◦ L i − d ◦ P i ) ◦ ( P i ◦ Γ i ◦ γ i ◦ t Γ i ◦ t P i ) = t P i . Because L = t L and γ i = t γ i , the second equality is the transpose of the first one. Thereforeit is enough to establish the first equality. Thanks to remark 2.16 we have Π j ◦ Γ i ◦ γ i = 0for all j < i and by transposing γ i ◦ t Γ i ◦ t Π j = 0 for all j < i . Expanding P i , we thereforesee that P i ◦ Γ i ◦ γ i ◦ t Γ i ◦ t P i = (1 −
12 Π d ) ◦· · ·◦ (1 −
12 Π i +1 ) ◦ Π i ◦ Γ i ◦ γ i ◦ t Γ i ◦ t Π i ◦ (1 − t Π i +1 ) ◦· · ·◦ (1 − t Π d ) . On the other hand, lemma 2.18 implies that t P i ◦ L i − d ◦ P i = (1 − t Π ) ◦ · · · ◦ (1 − t Π i +1 ) ◦ t Π i ◦ L i − d ◦ Π i ◦ (1 −
12 Π i − ) ◦ · · · ◦ (1 −
12 Π ) . Put altogether, this gives( P i ◦ Γ i ◦ γ i ◦ t Γ i ◦ t P i ) ◦ ( t P i ◦ L i − d ◦ P i ) =(1 −
12 Π d ) ◦ · · · ◦ (1 −
12 Π i +1 ) ◦ Π i ◦ Γ i ◦ γ i ◦ t Γ i ◦ t Π i ◦ L i − d ◦ Π i ◦ (1 −
12 Π i − ) ◦ · · · ◦ (1 −
12 Π ) . By proposition 2.4 if i is even and by proposition 2.8 if i is odd, we have Π i ◦ Γ i ◦ γ i ◦ t Γ i ◦ t Π i ◦ L i − d ◦ Π i = Π i . This finishes the proof of the Lefschetz isomorphism.Let’s now prove that the P i ’s satisfy the conclusion of lemma 2.18. A careful look atthe proof of lemma 2.18 shows that it is enough to show that P j factors through Γ j ◦ t γ j if Π j does. This can be read immediately from the formula defining Π j .If the projectors Π i factor through a 0-dimensional variety and if the projectors Π i +1 factor through a curve for all i , then it is clear from the formula that so will the pro-jectors P i and P i +1 . On the one hand, proposition 2.6 gives CH i ( X ) hom = Ker (cid:0) P i :19 H i ( X ) → CH i ( X ) (cid:1) and proposition 2.10 gives Ker (cid:0) AJ i : CH i ( X ) alg → J ai ( X ) ⊗ Q (cid:1) =Ker (cid:0) P i +1 : CH i ( X ) alg → CH i ( X ) alg (cid:1) . On the other hand, proposition 2.5 shows that(
X, P i ,
0) is isomorphic to ( L ⊗ i ) ⊕ d i and proposition 2.9 shows that ( X, P i +1 ,
0) is isomor-phic to h ( J ai ( X ))( i ). Alternately, the conclusion of theorem 2.13 gives these isomorphismsof Chow motives. Given a smooth projective complex variety X of dimension d , its i th Deligne cohomologygroup H D i ( X, Z ( p )) is the (2 d − i ) th hypercohomology group of the complex Z D ( d − p )given by 0 → Z · (2 iπ ) d − p −→ O X → Ω X → · · · → Ω d − p − X →
0. In other words, H D i ( X, Z ( p )) = H d − i ( X, Z D ( d − p )) . Deligne cohomology comes with a cycle map cl D i : CH Z i ( X ) → H D i ( X, Z ( i )) defined on theintegral Chow group CH Z i ( X ) which is functorial with respect to the action of correspon-dences and fits into an exact sequence0 → J i ( X ) → H D i ( X, Z ( i )) → Hdg i ( X ) → i ( X ) denotes the Hodge classes in H i ( X, Z ) and J i ( X ) is Griffiths’ i th interme-diate Jacobian. As proved in [5, Prop. 1], the following diagram with exact rows commutes0 / / CH Z i ( X ) hom AJ i (cid:15) (cid:15) / / CH Z i ( X ) cl D i (cid:15) (cid:15) / / CH Z i ( X ) / hom cl i (cid:15) (cid:15) / / / / J i ( X ) / / H D i ( X ) / / Hdg i ( X ) / / . (1)The homomorphism cl i : CH Z i ( X ) / hom → Hdg i ( X ) is always injective by definition of ho-mological equivalence. In particular the functoriality of the Deligne cycle class map impliesthe functoriality of the Abel-Jacobi map with respect to the action of correspondences. Lemma 3.1.
Let i be an integer such that d ≤ i ≤ d . • If the map cl : CH i ( X ) → H i ( X ) is surjective then H i ( X ) = N i H i ( X ) and H d − i ( X ) = N d − i H d − i ( X ) . • If the map cl D : CH i ( X ) → H D i ( X ) is surjective then H i +1 ( X ) = N i H i +1 ( X ) and H d − i − ( X ) = N d − i − H d − i − ( X ) . roof. If the map cl : CH i ( X ) → H i ( X ) is surjective then by definition H i ( X ) = N i H i ( X ). Because the Lefchetz isomorphism L i − d : H i ( X ) → H d − i ( X ) is inducedby a correspondence we also see that H d − i ( X ) = N d − i H d − i ( X ).Now suppose that the map cl D : CH i ( X ) → H D i ( X ) is surjective. A simple diagramchase in diagram 1 shows that the Abel-Jacobi map AJ i : CH i ( X ) hom → J i ( X ) ⊗ Q is then surjective. The Griffiths group Griff i ( X ) being countable, this is possible only if J i ( X ) ⊗ Q = J alg i ( X ) ⊗ Q . Therefore we have J i ( X ) ⊗ Q = J ai ( X ) ⊗ Q and hence H i +1 ( X ) = N i H i +1 ( X ). Again because the Lefchetz isomorphism L i − d +1 : H i +1 ( X ) → H d − i − ( X )is induced by a correspondence we also see that H d − i − ( X ) = N d − i − H d − i − ( X ). ⇒ intheorem 4 First we need a standard lemma.
Lemma 3.2.
Let N be a finite dimensional Chow motive. If its homology groups H ∗ ( N ) vanish then N = 0 .Proof. The homology class of id N ∈ End k ( N ) is then 0. Kimura [14, Prop. 7.2] provedthat if a Chow motive N is finite dimensional then the ideal of correspondences in End k ( N )which are homologically trivial is a nilpotent ideal. Hence id N is nilpotent i.e. id N = 0. Proof of ⇒ . Lemma 3.1 shows that H i ( X ) = N ⌊ i/ ⌋ H i ( X ) for all i . Therefore bylemma 2.1 the pairings N ⌊ i/ ⌋ H i ( X ) × N ⌊ (2 d − i ) / ⌋ H d − i ( X ) → Q are all non degenerate.Theorems 1, 2 and 3 then show that A := ⊕ h (Alb X ) ⊕ L ⊕ b ⊕ h ( J a ( X ))(1) ⊕ ( L ⊗ ) ⊕ b ⊕· · · ⊕ h ( J ad − ( X ))( d − ⊕ L ⊗ d is a direct summand of the Chow motive h ( X ) and that H ∗ ( A ) = H ∗ ( X ). Let’s write h ( X ) = A ⊕ N . The property of being finite dimensional isstable by direct summand. Therefore N is a finite dimensional motive. Moreover H ∗ ( N ) =0. Lemma 3.2 shows that N = 0. ⇔ in theorem 4 The results in this section are seemingly well-known. Given a smooth projective complexvariety X , we prove that the following statements are equivalent :1. CH i ( X ) alg is representable for all i .2. The total Abel-Jacobi map L i CH i ( X ) hom → L i J i ( X ) ⊗ Q is injective.3. The total Deligne cycle class map cl D : L i CH i ( X ) hom → L i H D i ( X, Q ( i )) is injec-tive.4. The total Deligne cycle class map is bijective and H i ( X ) = N ⌊ i/ ⌋ H i ( X ) for all i .21he equivalence (2 ⇔
3) follows immediately from diagram 1. The implication (4 ⇒ ⇒
4) is due to Esnault and Levine [6] (theorem 3.3below together with lemma 3.1). The main argument is a generalized decomposition ofthe diagonal as performed by Laterveer [16] and Paranjape [19] among others after Bloch’sand Srinivas’ original paper [4]. Proposition 3.4 proves the standard implication (2 ⇒ ⇒
2) so we include a proof of it, seecorollary 3.6. The proof goes through a generalized decomposition of the diagonal as donein [6, Theorem 1.2] with some minor changes (theorem 3.5).
Theorem 3.3 (Esnault-Levine) . Let s be an integer. Assume that the rational Delignecycle class maps cl D i : CH i ( X ) → H D i ( X ) are injective for all i ≤ s . Then these areall surjective. Moreover the rational cycle class maps cl i : CH i ( X ) → H i ( X ) are alsosurjective for all i ≤ s .Proof. The fact that the rational Deligne cycle class maps are surjective for all i ≤ s is contained in Theorem 2.5 of [6] (the maps cl D i are denoted cl d − i , in [6]). The claimabout the rational cycle class maps being surjective is Corollary 2.6 (which states that N i H i ( X ) = Hdg i ( X ) ⊗ Q ) together with Theorem 3.2 (which states in particular that H i ( X ) = Hdg i ( X ) ⊗ Q ) of [6]. Proposition 3.4.
Given i , if the Abel-Jacobi map CH i ( X ) alg → J i ( X ) ⊗ Q is injective,then CH i ( X ) alg is representable.Proof. Let J ai ( X ) be the image of the Abel-Jacobi map AJ i : CH Z i ( X ) alg → J i ( X ). Bydefinition of algebraic equivalence, CH i ( X ) alg := P Im (cid:0) Γ ∗ : CH ( C ) hom → CH i ( X ) (cid:1) where the sum runs through all smooth projective curves C and all correspondences Γ ∈ CH i +1 ( C × X ). Therefore, by functoriality of the Abel-Jacobi map, we have J ai ( X ) = P Im (cid:0) Γ ∗ : J ( C ) → J ai ( X ) (cid:1) . By finiteness properties of abelian varieties there exist acurve and a correspondence Γ ∈ CH i +1 ( C × X ) such that J ai ( X ) = Γ ∗ J ( C ). Therefore forthis particular curve Γ ∗ CH ( C ) hom = CH i ( X ) alg . Theorem 3.5.
Let s be an integer with ≤ s ≤ d and let X be a d -dimensional smoothprojective complex variety. Assume CH i ( X ) alg is representable for all i ≤ s . Then, thereis a decomposition ∆ X = γ + γ + · · · + γ s + γ s +1 ∈ CH d ( X × X ) such that γ i is supported on D i × Γ i +1 for some subschemes D i and Γ i +1 of X satisfying dim D i = d − i and dim Γ i +1 = i +1 and γ s +1 is supported on D s +1 × X for some subscheme D s +1 of X satisfying dim D s +1 = d − s − .Proof. The proof is the same as the proof of [6, lemma 1.1] once one remarks that the map ch : CH ( ˜ D ) → CH n ( X ) on page 207 has image contained in CH n ( X ) alg and thereforefactors through the Albanese map CH ( ˜ D ) → Alb ˜ D , because CH n ( X ) alg is representableand has thus the structure of an abelian variety.22 orollary 3.6. Assume CH i ( X ) alg is representable for all i ≤ s . Then the Abel-Jacobimaps AJ i : CH i ( X ) hom → J i ( X ) ⊗ Q are injective for all i ≤ s .Proof. By assumption made on CH ∗ ( X ) alg , the diagonal ∆ X admits a decomposition asin theorem 3.5. For all i ≤ s , let e Γ i +1 be a desingularization of Γ i +1 . The action ofthe correspondence γ i on CH j ( X ) hom then factors through CH j ( e Γ i +1 ) hom . For dimensionreasons γ i acts possibly non trivially only on CH i ( X ) and CH i +1 ( X ). Also for dimensionreasons, the correspondence γ s +1 acts trivially on CH i ( X ) for i ≤ s . Therefore the cycle γ i acts non trivially on CH j ( X ) hom only if i = j . Thus the action of γ i on CH i ( X ) hom isidentity. Finally, by functoriality of the algebraic Abel-Jacobi map, we have the followingcommutative diagram for all i ≤ sCH i ( X ) hom AJ i (cid:15) (cid:15) / / CH i ( e Γ i +1 ) hom ≃ (cid:15) (cid:15) / / CH i ( X ) hom AJ i (cid:15) (cid:15) J i ( X ) / / J i ( e Γ i +1 ) / / J i ( X ) . The composition of the two maps on each row is induced by γ i and is equal to identity up totorsion. A diagram chase then shows that AJ i : CH i ( X ) alg → J ai ( X ) ⊗ Q is injective. Remark 3.7.
Given i , I cannot prove that if CH i ( X ) alg is representable then the Abel-Jacobi map AJ i : CH i ( X ) alg → J i ( X ) ⊗ Q restricted to algebraically trivial cycles isinjective. Remark 3.8.
Bloch and Srinivas proved [4, Theorem 1(i)] that if CH ( X ) alg is rep-resentable then so is CH ( X ) alg . A generalized decomposition of the diagonal showsthat if CH ( X ) alg , . . . , CH s ( X ) alg are representable then CH ( X ) alg , . . . , CH s ( X ) alg are also representable. Therefore, if d is the dimension of X , it is enough to know that CH ( X ) alg , . . . , CH ⌊ d/ ⌋− ( X ) alg are representable in order to deduce that CH ∗ ( X ) alg isrepresentable. ⇒ intheorem 4 In order to prove the implication 4 ⇒ i,i and Π i +1 ,i , together with the following lemma which appears in [8, lemma 1]. Lemma 3.9.
Let N be a Chow motive over a field k and let Ω be a universal domain over k , i.e. an algebraically closed field of infinite transcendence degree over k . If CH ∗ ( N Ω ) = 0 ,then N = 0 .Proof of ⇒ . If CH ∗ ( X C ) alg is representable then corollary 3.6 shows that the Delignecycle class maps cl D i are all injective. By Esnault and Levine’s theorem 3.3, the Delignecycle class maps cl D i and the cycle class maps cl i are surjective for all i . Now lemma 3.123hows that H i ( X ) = N i H i ( X ) and H i +1 ( X ) = N i H i +1 ( X ) for all i . Thanks to lemma2.1 we can therefore apply theorems 1, 2 and 3 to cut out the motive ⊕ h (Alb X ) ⊕ L ⊕ b ⊕ h ( J a ( X ))(1) ⊕ ( L ⊗ ) ⊕ b ⊕ · · · ⊕ h ( J ad − ( X ))( d − ⊕ L ⊗ d from h ( X ). These two motiveshave same rational Chow groups when the base field is extended to C , lemma 3.9 impliesthey are equal.As a corollary, we obtain a result proved independently by Kimura [15] (Kimura’s resultworks more generally for any pure Chow motive over C ). Proposition 3.10.
Let X be a d -dimensional smooth projective variety over k . If CH ∗ ( X C ) is a finite dimensional Q -vector space, then h ( X ) ≃ d M i =0 ( L ⊗ i ) ⊕ b i . Moreover, the cycle class maps cl i : CH i ( X ) → H i ( X ) are all isomorphismsProof. Indeed, if CH ∗ ( X C ) is a finite dimensional Q -vector space then it is representable.Apply theorem 4 to see that h ( X ) is a direct sum of Lefschetz motives and twisted h ’s ofabelian varieties. Now for a complex abelian variety J , CH ( h ( J )) = J ⊗ Q which is aninfinite dimensional Q -vector space if J = 0. Therefore h ( X ) is a direct sum of Lefschetzmotives only. A smooth projective variety X of dimension d is said to have a Chow-K¨unneth decom-position (CK decomposition for short) if there exist mutually orthogonal idempotentsΠ , Π , . . . Π d ∈ CH d ( X × X ) adding to the identity ∆ X such that (Π i ) ∗ H ∗ ( X ) = H i ( X )for all i . In this section, we wish to give explicit examples of varieties having a Chow-K¨unneth decomposition. For this purpose we use the projectors of theorems 1 and 2.Along the way we are able to establish Grothendieck’s standard conjectures and Kimura’sfinite dimensionality conjecture in some new cases. In [21], we prove Murre’s conjecturesfor all the varieties considered in §§ Here, X denotes a smooth projective variety of dimension d . All the varieties for whichwe will be able to show that they admit a CK decomposition will also satisfy the motivicLefschetz conjecture. Definition 4.1.
The variety X is said to satisfy the motivic Lefschetz conjecture if itadmits a CK decomposition { Π i } ≤ i ≤ d such that, for all i > d the morphism of Chowmotives ( X, Π i , → ( X, Π d − i , i − d ) given by intersecting i − d times with a hyperplanesection is an isomorphism. 24t is immediate to see that if X satisfies the motivic Lefschetz conjecture then it satisfiesthe Lefschetz standard conjecture. Since Grothendieck’s standard conjectures for X reduceto the standard Lefschetz conjecture for X in characteristic zero [1, 5.4.2.2], we get that X satisfies all of Grothendieck’s standard conjectures.The key result of this section is the following. Theorem 4.2. If H i ( X ) = N ⌊ i/ ⌋ H i ( X ) for all i > d , then X has a Chow-K¨unnethdecomposition { P i } ≤ i ≤ d where the idempotents P i for i = d satisfy all the propertieslisted in theorems 1 and 2. In particular the motivic Lefschetz conjecture holds for X andhence, also do the standard conjectures hold for X .Proof. If H i ( X ) = N ⌊ i/ ⌋ H i ( X ) for all i > d , then by Poincar´e duality the pairings N ⌊ i ⌋ H i ( X ) × N ⌊ d − i ⌋ H d − i ( X ) → Q are non degenerate for all i > d . The motivic Lef-schetz isomorphisms of theorems 1 and 2 then imply that X satisfies the Lefschetz standardconjecture. Therefore, by lemma 2.1, the pairing N ⌊ d/ ⌋ H d ( X ) × N ⌊ d/ ⌋ H d ( X ) → Q is alsonon degenerate. By theorem 3, we get mutually orthogonal idempotents { Π i } ≤ i ≤ d . Let’sput P i := Π i for i = d and P d := ∆ X − P i = d Π i . Then { P i } ≤ i ≤ d is the required CKdecomposition for X . An immediate consequence to theorem 4.2 is the following.
Corollary 4.3.
Let Y be a -fold with H ( Y, O Y ) = 0 , e.g. a Calabi-Yau -fold. Then Y has a CK decomposition and the motivic Lefschetz conjecture holds for Y .Proof. By the Lefschetz (1 , H ( Y, O Y ) = 0 implies H ( Y ) = N H ( Y ). Thus H i ( X ) = N ⌊ i/ ⌋ H i ( X ) for all i > X has a CK decom-position, we give some information on the support of the middle CK projector of X . Suchinformation will be used in [21] to further prove Murre’s conjectures in the cases coveredby the theorems. For this purpose we need the following lemma that appears in [22, lemma1.2] and which is essentially due to Kahn and Sujatha [12]. Lemma 4.4.
Let M = ( X, p, be a Chow motive over k and let Ω be a universal domainover k . If CH ( N Ω ) = 0 , then there exists a Chow motive N = ( Y, q, such that M =( Y, q, . Theorem 4.5.
Let X be a smooth projective variety of even dimension d = 2 n . If CH ( X ) alg , CH ( X ) alg , . . . , CH n − ( X ) alg are representable, then X has a CK decompo-sition { Π i } and the motivic Lefschetz conjecture holds for X . Moreover, the idempotents Π i are as in theorems 1 and 2 for i = d and the idempotent Π d has a representativesupported on X × Z with Z a subvariety of X of dimension n + 1 . roof. By a generalized decomposition of the diagonal (as performed for instance by Later-veer [16, 2.1]), the assumption on the Chow groups of X implies that H i ( X ) = N ⌊ i/ ⌋ H i ( X )for all i > d . We can therefore apply theorem 4.2 to get a CK decomposition { Π i } ≤ i ≤ d for X where the idempotents Π i for i = d satisfy all the properties listed in theorems 1 and2. By corollary 3.6 if CH ( X ) alg , CH ( X ) alg , . . . , CH n − ( X ) alg are representable, then theAbel-Jacobi maps AJ i : CH i ( X ) hom → J i ( X ) ⊗ Q are injective for all i ≤ n −
2. Esnaultand Levine’s theorem 3.3 then implies that the Abel-Jacobi maps are bijective. Thanksto the properties of the CK projectors, we thus get CH i ( X ) = (Π i + Π i +1 ) ∗ CH i ( X ) forall i ≤ n −
2. As such, the idempotent Π d acts trivially on CH i ( X ) for all i ≤ n −
2. Byapplying n − X, Π d ,
0) is isomorphic to some Chow motive(
Y, q, n − f ∈ Hom(( X, Π d , , ( Y, q, n − d = Π d ◦ f − ◦ q ◦ f ◦ Π d . In particular Π d factors through Y and a straightfor-ward analysis of the dimensions shows that Π d has a representative supported on X × Z with Z a subvariety of X of dimension n + 1. Corollary 4.6.
Every fourfold X with CH ( X ) alg representable has a CK decompositionand satisfies the motivic Lefschetz conjecture and hence the standard conjectures. Corollary 4.7.
Let X be a smooth projective fourfold admitting a curve C as a base forits maximal rationally connected fibration. This means that there exists a rational map f : X C with rationally connected general fiber. Then X has a CK decomposition andsatisfies the motivic Lefschetz conjecture and hence the standard conjectures.Proof. CH ( X ) alg is representable. Corollary 4.8.
Every rationally connected smooth projective fourfold (e.g. every smoothprojective variety which is birational to a Fano fourfold) has a Chow-K¨unneth decompositionand satisfies the motivic Lefschetz conjecture.Proof.
A rationally connected smooth projective fourfold X satisfies CH ( X ) = Q . Remark 4.9.
Arapura [3] proved the Lefschetz standard conjecture for unirational four-folds. He does so by proving that a unirational fourfold is motivated by surfaces. Moregenerally, Arapura proves that any variety which is motivated by a surface (this meansthat the cohomology of X is generated by the cohomology of product of surfaces via cor-respondences) satisfies the standard Lefschetz conjecture.Corollary 4.6 is more precise for unirational fourfolds because we obtain the Lefschetzisomorphism modulo rational equivalence (rather than just modulo homological equiva-lence). Moreover, corollary 4.6 includes the case of rationally connected fourfolds as wellas the case of fourfolds admitting a curve as a base for their maximal rationally connectedfibration.Let’s also mention that in what follows Arapura’s technique doesn’t seem to apply toprove the Lefschetz standard conjecture because the middle cohomology of the varieties inquestion is not necessarily generated by the cohomology of products of surfaces.26 heorem 4.10. Let X be a smooth projective variety of odd dimension d = 2 n + 1 with H n +1 ( X, Ω n − X ) = 0 . If CH ( X ) alg , CH ( X ) alg , . . . , CH n − ( X ) alg are representable, then X has a CK decomposition { Π i } and the motivic Lefschetz conjecture holds for X . More-over, the idempotents Π i are as in theorems 1 and 2 for i = d and the idempotent Π d hasa representative supported on X × Z with Z a subvariety of X of dimension n + 2 .Proof. As for the proof of theorem 4.5, a generalized decomposition of the diagonal ar-gument shows that the assumption on the Chow groups of X implies that H i ( X ) = N ⌊ i/ ⌋ H i ( X ) for i > d + 1 and that H d +1 ( X ) = N d − H d +1 ( X ). This last equality meansthat there is a smooth projective variety S of dimension n + 2 and a map f : S → X such that f ∗ H ( S ) = H d +1 ( X ). Because H n +1 ( X, Ω n − X ) = 0, we see that H d +1 ( X ) ismade of Hodge classes. By the Lefschetz (1 , S , we see that H d +1 ( X )is spanned by algebraic cycles, i.e. that H d +1 ( X ) = N d +12 H d +1 ( X ). We can thus applytheorem 4.2 to get a CK decomposition { Π i } ≤ i ≤ d that satisfies the motivic Lefschetztheorem. The proof of the fact that Π d has a representative supported on X × Z for Z a subvariety of X of dimension n + 2 goes along the same lines as the proof of theorem4.5. Corollary 4.11.
Let X be a smooth projective fivefold. If CH ( X ) alg is representable andif H ( X, Ω X ) = 0 , then X has a CK decomposition and satisfies the standard conjectures. Corollary 4.12.
Let X be a smooth projective rationally connected fivefold, e.g a fivefoldwhich is birational to a Fano fivefold. If H ( X, Ω X ) = 0 , then X has a CK decompositionand satisfies the standard conjectures. Otwinowska [18] proved that if X is a smooth hyperplane section of a hypersurface in P n +1 covered by l -planes then CH i ( X ) hom = 0 for i ≤ l − l = ⌊ n/ ⌋ the Chow groups CH i ( X ) alg are all representableby remark 3.8. As a direct application of theorem 4 we get Theorem 4.13.
Let l = ⌊ n/ ⌋ and let X be a smooth hyperplane section of a hypersurfacein P n +1 covered by l -planes. Then, • if n − is even, h ( X ) = ⊕ L ⊕ L ⊗ ⊕ · · · ⊕ L ⊗ n − . • if n − is odd, h ( X ) = ⊕ L ⊕ · · · ⊕ L ⊗ l ⊕ h ( J alg l )( l ) ⊕ L ⊗ l +1 ⊕ · · · ⊕ L ⊗ n − .Moreover, in any case, h ( X ) is finite dimensional in the sense of Kimura. Remark 4.14.
Otwinowska also mentions that if k ( n − l ) − (cid:0) d + ld (cid:1) + 1 ≥
0, any smooth pro-jective hypersurface of degree d in P n C is covered by linear projective varieties of dimension l . 27 xamples 4.15. Here are some varieties for which theorem 4 and the results of [7] makeit possible to prove that they have finite dimensional Chow motive : • Cubic 5-folds. • A 5-fold which is the smooth intersection of a cubic and a quadric. • A 7-fold which is the smooth intersection of two quadrics.Further examples of varieties with finite dimensional Chow motive can be constructed asfollows. Let X be a variety as in the theorem above. Consider smooth projective varietiesobtained from X by successively blowing up smooth curves. Then, by the blowing-upformula for Chow motives, such varieties have finite dimensional Chow motive. Moreoverany variety Y which is dominated by a product of such varieties has finite dimensionalChow motive. References [1] Yves Andr´e.
Une introduction aux motifs (motifs purs, motifs mixtes, p´eriodes) , volume 17 of
Panora-mas et Synth`eses [Panoramas and Syntheses] . Soci´et´e Math´ematique de France, Paris, 2004.[2] Yves Andr´e and Bruno Kahn. (with an appendix by P. O’Sullivan) Nilpotence, radicaux et structuresmono¨ıdales.
Rend. Sem. Mat. Univ. Padova , 108:107–291, 2002.[3] Donu Arapura. Motivation for Hodge cycles.
Adv. Math. , 207(2):762–781, 2006.[4] S. Bloch and V. Srinivas. Remarks on correspondences and algebraic cycles.
Amer. J. Math. ,105(5):1235–1253, 1983.[5] Fouad El Zein and Steven Zucker. Extendability of normal functions associated to algebraic cycles.In
Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) , volume 106 of
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