aa r X i v : . [ m a t h . AG ] A p r SMASH NILPOTENCE ON UNIRULED 3-FOLDS
RONNIE SEBASTIAN
Abstract.
Voevodsky has conjectured that numerical and smash equiv-alence coincide on a smooth projective variety. We prove this conjectureholds for uniruled 3-folds and for one dimensional cycles on products ofKummer surfaces. Introduction
Throughout this article we work over an algebraically closed field k andwith algebraic cycles with rational coefficients.Let X be a smooth and projective variety over k . In [Voe95], Voevodskydefines a cycle α to be smash nilpotent if the cycle α n := α × α . . . × α on thevariety X n := X × X . . . × X is rationally equivalent to 0. It is trivial to seethat a smash nilpotent cycle is numerically trivial, Voevodsky conjecturedthat the converse also holds. Voevodsky, [Voe95], and Voisin, [Voi94], provethat a cycle which is algebraically trivial is smash nilpotent.Kimura, [Kim05, Proposition 6.1], proved that a morphism between fi-nite dimensional motives of different parity is smash nilpotent. Thus, if analgebraic cycle can be viewed as a morphism between motives of differentparities, then it is smash nilpotent. In [KS09], the authors use this fact toprove that skew cycles on an abelian variety are smash nilpotent. A cy-cle β is called skew if it satisfies [ − ∗ β = − β . In [KS09] such cycles areexpressed as morphisms between motives of different parity, using the factthat the motive of an abelian variety has a Chow-Kunneth decomposition, h ( A ) = A M i =0 h i ( A )and the motives h i ( A ) for i odd are oddly finite dimensional.In [Seb13] it is proved that for one dimensional cycles on a variety domi-nated by a product of curves, smash equivalence and numerical equivalencecoincide. The same result can be deduced from [Mar08] and [Her07], whereit is shown that for a smooth projective curve C , for any adequate equiv-alence relation, [ C ] i = 0 implies that [ C ] i +1 = 0, for i ≥
2. Here [ C ] i denotes the Beauville component of the curve C in its Jacobian satisfying[ n ] ∗ [ C ] i = n i [ C ] i . If we combine this with [KS09], where it is shown that Mathematics Subject Classification.
Key words and phrases.
Algebraic cycles, smash nilpotence. [ C ] = 0 modulo smash equivalence, then one can deduce the results in[Seb13].If we take the Chow ring of an abelian variety modulo algebraic equiva-lence and go modulo the subring generated by the cycles in the precedingparagraphs under the Pontryagin product, intersection product and Fouriertransform, then there are no nontrivial examples of higher dimensional cycles(dim >
1) for which Voevodsky’s conjecture holds.The purpose of this article is to write down some more examples for whichthis conjecture holds. The main theorems in this article are
Theorem 1.
Let X be uniruled 3-fold. Then numerical and smash equiva-lence coincide for cycles on X . Theorem 2.
Let K i , i = 1 , , . . . , N be Kummer surfaces. Then numericaland smash equivalence coincide for one dimensional cycles on X := K × K × · · · × K N . The proof of the above theorems use Lemma 3, which implies the follow-ing. If numerical and smash equivalence coincide on a smooth and projectivevariety Y , then they coincide on ˜ Y , which is obtained by blowing up Y alonga smooth subvariety of dimension ≤ Acknowledgements . We thank Najmuddin Fakhruddin for useful dis-cussions. 2.
Smash equivalence and blow ups
Let Y be a smooth variety and i : X ֒ → Y be a smooth and closedsubvariety. Let f : ˜ Y → Y denote the blow-up of Y along X . Lemma 3.
If numerical and smash equivalence coincide for elements in CH i ( X ) for i ≤ r and CH r ( Y ) , then they coincide for elements in CH r ( ˜ Y ) .Proof. Consider the Cartesian square˜ X / / g (cid:15) (cid:15) ˜ Y f (cid:15) (cid:15) X i / / Y Then [Ful97, Proposition 6.7] says that there is an exact sequence0 → CH r ( X ) → CH r ( ˜ X ) ⊕ CH r ( Y ) → CH r ( ˜ Y ) → X is a smooth subvariety of Y , we have that ˜ X g −→ X is the projectivebundle associated to the locally free sheaf I X / I X on X . Thus, everyelement β ∈ CH r ( ˜ X ) may be expressed as the sum β = d − X i =0 c ( O (1)) i ∩ g ∗ g ∗ ( β ∩ c ( O (1)) d − − i ) , MASH NILPOTENCE ON UNIRULED 3-FOLDS 3 where O (1) is the tautological bundle on ˜ X . If we assume that numericallytrivial elements in CH i ( X ) are smash nilpotent for i ≤ r , then the aboveformula shows that numerically trivial elements in CH r ( ˜ X ) are smash nilpo-tent. If numerical and smash equivalence coincide for elements in CH r ( Y ),then the above exact sequence would show that these coincide for elementsin ˜ Y as well. (cid:3) The following is a standard result which we include for the benefit of thereader.
Lemma 4.
Let X be a smooth projective variety and let h : Y → X be adominant morphism. If numerical and smash equivalence coincide for cycleson Y , then they coincide for cycles on X .Proof. Let l ∈ CH ( Y ) be a relatively ample line bundle. The relativedimension of h is r := dim(Y) − dim(X) and define d by h ∗ ( l r ) =: d [ Y ].Then by the projection formula, we have ∀ α ∈ CH ∗ ( X ) h ∗ ( l r · h ∗ α ) = dα If α is a numerically trivial cycle on X , then l r · h ∗ α is a numerically trivialcycle on Y and so is smash nilpotent. The above equation shows that α issmash nilpotent. (cid:3) Examples
Uniruled 3-folds.Definition 5.
By a uniruled 3-fold we mean a smooth projective variety X for which there is a dominant rational map ϕ : S × P X for somesmooth projective surface S . Proof of Theorem 1.
Since X is projective and Y := S × P is normal, ϕ can be defined on an open set U whose compliment has codimension ≥
2. Let
X ֒ → P n be a closed immersion, composing this with ϕ we get a morphism g : U → P n . Let L denote the pullback of O (1) along g . Since Y \ U hascodimension ≥
2, there is a unique line bundle on Y which restricts to L ,we denote this also by L . As Y is smooth and codimension Y \ U is ≥
2, therestriction map H ( Y, L ) → H ( U, L ) is an isomorphism, see, for example[Har77, Chapter 3, Ex 3.5]. Let V ⊂ H ( Y, L ) be the subspace of globalsections g ∗ H ( P n , O (1)). Let J ⊂ L be the subsheaf generated by V , then I = J ⊗ L − is an ideal sheaf such that Y \ Supp(I) = U and V is containedin the image of the map H ( Y, I ⊗ L ) → H ( Y, L ) and it generates I ⊗ L .We want to apply the principalization theorem to the ideal sheaf I . Incharacteristic 0, see [Kol07, Theorem 3.21], and in positive characteristic, see[Cut09, Theorem 1.3]. We get a morphism f : Y ′ → Y which is obtained as acomposite of smooth blow-ups, such that f ∗ I is a locally principal ideal sheafand f is an isomorphism on f − ( U ). The subspace f ∗ V ⊂ H ( Y ′ , f ∗ I ⊗ f ∗ L )defines a map Y ′ → P n which extends g . Thus, we get a dominant morphism R. SEBASTIAN Y ′ → X . As S is a surface, numerical and smash equivalence coincide forcycles on S and so for cycles on Y . Since Y ′ is obtained from Y by blowingup at smooth centers and dim(Y) = 3, numerical and smash equivalencecoincide for Y ′ using Lemma 3. Finally, use Lemma 4 to get the same resultfor X . (cid:3) Kummer surfaces.
Let Y be an abelian surface and let X be the setof 2 torsion points. These are exactly the fixed points for the involution x x − on Y . This involution lifts to an involution of ˜ Y which we denote˜ i and the quotient ˜ Y / ˜ i is the Kummer surface associated to Y . We denotethis surface by K and by π the quotient map ˜ Y → K .Let Y i , i = 1 , K i , i = 1 , X i be the set of 2 torsion points in Y i . Similarly,we have the varieties ˜ Y i and there is a dominant projective map ˜ Y × ˜ Y → K × K .The map ˜ Y × ˜ Y → Y × Y may be factored as the composite of two blowups ˜ Y × ˜ Y → ˜ Y × Y → Y × Y the first along the surface X × Y and the second along the surface ˜ Y × X .Applying Lemma 3 to both these blow ups, we get that numerical andsmash equivalence coincide for one dimensional cycles on ˜ Y × ˜ Y and sousing Lemma 4 they coincide on K × K . Proof of Theorem 2.
We recall a result from [Seb13] which we need. Let N ≥ C be a smooth projective curve with a basepoint c . Let ∆ C denote the diagonal embedding C ֒ → C N . Let p ij : C N → C N denote the map which leaves the i th and j th coordinates intact andthe other coordinates are changed to c , for example, p ( x , x , . . . , x N ) =( x , x , c , c , . . . , c ). Then there are rational numbers q ij such that(3.1) ∆ C ∼ sm X i = j q ij p ij ∗ (∆ C ) . Let X := K × K . . . × K N be a product of Kummer surfaces. Fix basepoints e i ∈ K i , and define (we abuse notation here) p ij : X → X in thesame way as above, using these base points. We remark that if we workmodulo algebraic equivalence, for any cycle α ∈ CH ∗ ( X ), the cycle p ij ∗ ( α )is independent of the choice of these base points. Hence, the same is truemodulo smash equivalence.Let D ֒ → X be a reduced and irreducible one dimensional subvariety. Let C → D be its normalization and denote the composite map by f : C → X .If we let p i denote the projection from X to K i and let π := ( p ◦ f ) × ( p ◦ f ) . . . ( p n ◦ f ) , MASH NILPOTENCE ON UNIRULED 3-FOLDS 5 then we get f ∗ ([ C ]) = π ∗ (∆ C ). Using equation (3.1), we get that modulosmash equivalence[ D ] = f ∗ ([ C ]) = π ∗ (∆ C ) = X i = j q ij π ∗ p ij ∗ (∆ C ) = X i = j q ij p ij ∗ π ∗ (∆ C )= X i = j q ij p ij ∗ ([ D ])In particular, we get for any one dimensional cycle α , α = X i = j q ij p ij ∗ ( α )As we have seen above, on a product of two Kummer surfaces, numerical andsmash equivalence coincide for one dimensional cycles. If α is numericallytrivial, each p ij ∗ ( α ) is numerically trivial and so smash nilpotent. Thus, α is smash nilpotent. (cid:3) References [Cut09] Cutkosky,
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