On a question of O'Grady about modified diagonals
aa r X i v : . [ m a t h . AG ] N ov ON A QUESTION OF O’GRADY ABOUT MODIFIED DIAGONALS by Ben Moonen and Qizheng Yin
Abstract.
Let X be an abelian variety of dimension g . In a recent preprint O’Grady defines modified diagonalclasses Γ m on X m and he conjectures that the class of Γ m in the Chow ring of X m is torsion for m > g + 1.We prove a generalization of this conjecture. AMS 2010 Mathematics subject classification: Throughout this note, R is a Dedekind ring and S is a connected scheme that is smooth andof finite type over R , which is the base scheme over which we work.Let π : Y → S be a smooth projective S -scheme, a : S → Y a section of π , and m > Y m we mean the product of m copies of Y relative to S . For j ⊂ { , . . . , m } letpr j : Y m → Y be the projection onto the j th factor and let ˆpr j : Y m → Y m − be the projectionthat contracts the j th factor.For I ⊂ { , . . . , m } , let δ I : Y → Y m be the morphism defined by the property that pr i ◦ δ I is the identity on Y if i ∈ I and is the “constant map” a ◦ π if i / ∈ I . Following O’Grady [3] wedefine an algebraic cycle Γ m ( Y, a ) on Y m byΓ m ( Y, a ) = X ∅6 = I ⊂{ ,...,m } ( − m −| I | · δ I, ∗ ( Y ) . The goal of this note is to prove the following result, which proves Conjecture 0.3 of [3],generalized to the relative setting. Theorem. — Let X → S be an abelian scheme of relative dimension g . For any section a ∈ X ( S ) the class of Γ m ( X, a ) in CH( X m ) is torsion if m > g + 1 . Remark.
Still with X an abelian scheme over S , let e ∈ X ( S ) be the zero section. Translationover a gives an isomorphism t a : X ∼ −→ X over S . The class of Γ m ( X, a ) is the push-forward ofthe class of Γ m ( X, e ) under t ma : X m ∼ −→ X m . Hence it suffices to prove the theorem taking e assection. Lemma. — Let
X/S be an abelian scheme of relative dimension g and write Γ m = Γ m ( X, e ) .For n ∈ Z let mult X ( n ): X → X be the endomorphism given by multiplication by n . (i) For n ∈ Z we have mult X m ( n ) ∗ [Γ m ] = n g · [Γ m ] . (ii) For j ∈ { , . . . , m } let ˆpr j : X m → X m − be the projection map contracting the j th factor.Then ˆpr j, ∗ [Γ m ] = 0 .Proof. For (i) we use that, with notation as in point 1, and taking a = e as section, mult X m ( n ) ◦ δ I is the same as δ I ◦ mult X ( n ) and that mult X ( n ) is finite flat of degree n g .For (ii), consider a non-empty subset I ⊂ { , . . . , m } . If I = { j } then ˆpr j ◦ δ I : X → X m − isconstant, so that ˆpr j, ∗ δ I, ∗ ( X ) = 0. Let I be the set of non-empty subsets of { , . . . , m } different1rom { j } . Then I = I ` I where I ∈ I if j / ∈ I and I ∈ I if j ∈ I . The map β : I → I given by I I ∪ { j } is a bijection. Further, for I ∈ I we have ˆpr j ◦ δ I = ˆpr j ◦ δ β ( I ) and | β ( I ) | = | I | + 1. In the calculation of ˆpr j, ∗ [Γ m ], the terms corresponding to I and β ( I ) thereforecancel, and this gives the assertion. (cid:3) Let
Mot S be the category of Chow motives over S with respect to graded correspondences;see for instance [2], Section 1. If Y is a smooth projective S -scheme we write h ( Y ) for its Chowmotive. In Mot S we have a tensor product with h ( Y ) ⊗ h ( Z ) = h ( Y × S Z ).Let S = h ( S ) be the unit motive and ( n ) the n th Tate twist. If M is a motive, wewrite M ( n ) = M ⊗ ( n ). The Chow groups (with Q -coefficients) of a motive M are defined byCH i ( M ) Q = Hom Mot S (cid:0) ( − i ) , M (cid:1) .If f : Y → Z is a morphism of smooth projective S -schemes we have induced morphisms f ∗ : h ( Z ) → h ( Y ) and, assuming Y and Z are connected, f ∗ : h ( Y ) → h ( Z ) (cid:0) d (cid:1) , where d =dim( Z/S ) − dim( Y /S ). Let
X/S be an abelian scheme of relative dimension g . As proven by Deninger and Murrein [2] (generalizing results of Beauville [1] over a field) we have a canonical decomposition h ( X ) = ⊕ gi =0 h i ( X ) in Mot S that is stable under all endomorphisms mult X ( n ) ∗ , and such thatmult X ( n ) ∗ is multiplication by n g − i on h i ( X ). For m > h ( X m ) = M i =( i ,...,i m ) m O j =1 h i j ( X ) , where the sum runs over the elements i ∈ { , . . . , g } m . Under this decomposition we have(6.1) h ν ( X m ) = M | i | = ν m O j =1 h i j ( X ) , where the sum runs over the m -tuples i = ( i , . . . , i m ) in { , . . . , g } m with | i | = i + · · · + i m equal to ν .If π : X → S is the structural morphism, π ∗ : h ( X ) → h ( S ) (cid:0) − g (cid:1) = ( − g ) is an isomorphismon h g ( X ) and is zero on ⊕ g − i =0 h i ( X ) Lemma. — Notation as above. If there is an index ν such that i ν = 2 g then the componentof (cid:2) Γ m ( X, e ) (cid:3) in CH( ⊗ mj =1 h i j ( X )) Q is zero.Proof. Assume i ν = 2 g . Consider the projection ˆpr ν : X m → X m − . By the fact stated justbefore the lemma, the induced map ˆpr ν, ∗ : h ( X m ) → h ( X m − ) (cid:0) − g (cid:1) restricts to an isomor-phism of ⊗ mj =1 h i j ( X ) with a sub-motive of h ( X m − ) (cid:0) − g (cid:1) . The assertion therefore follows fromLemma 4(ii). (cid:3) Proof of the Theorem.
Write Γ m = Γ m ( X, e ). By Lemma 4(i) we have[Γ m ] ∈ CH (cid:0) h g ( m − ( X m ) (cid:1) Q ⊂ CH( X m ) Q , and (6.1) gives CH (cid:0) h g ( m − ( X m ) (cid:1) Q = M | i | =2 g ( m − CH( ⊗ mj =1 h i j ( X )) Q . m > g + 1 implies that for every i = ( i , . . . , i m ) ∈ { , . . . , g } m with | i | = 2 g ( m −
1) there is an index ν with i ν = 2 g and by Lemma 7 we are done. (cid:3) References [1] A. Beauville,
Sur l’anneau de Chow d’une vari´et´e ab´elienne.
Math. Ann. 273 (1986), 647–651.[2] C. Deninger, J. Murre,
Motivic decomposition of abelian schemes and the Fourier transform.
J. reineangew. Math. 422 (1991), 201–219.[3] K.G. O’Grady,
Computations with modified diagonals . Preprint, arXiv:1311.0757v1.Radboud University Nijmegen, IMAPP, PO Box 9010, 6500GL Nijmegen, The [email protected] [email protected]. Preprint, arXiv:1311.0757v1.Radboud University Nijmegen, IMAPP, PO Box 9010, 6500GL Nijmegen, The [email protected] [email protected]