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A note on the Picard motive of a variety
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Indagationes Mathematicae 23 (2012) 377–380 www.elsevier.com/locate/indag
A note on the Picard motive of a variety Mustafa Devrim Kaba Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Nijenborgh 9, 9747 AG, Groningen, The Netherlands Received 9 May 2010; received in revised form 27 January 2012; accepted 28 January 2012
Communicated by H.W. Broer
Abstract We prove that the Picard motive of a smooth projective variety and the Picard motive of its Albanese variety are isomorphic, under the assumption that both the variety and its Albanese variety have dimension at least 2. c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Chow motives; Picard motive; Albanese variety
1. Introduction For various Weil cohomology theories (H ∗ (·), e.g., singular cohomology, e´ tale cohomology, etc.), we have the well-known isomorphism H 1 (X ) ≃ H 1 (J X ), where X is a smooth projective variety and J X is its Albanese variety, both defined over some field k. Therefore, it is natural to ask if a similar isomorphism holds in the category of pure Chow motives (M). More precisely, if we let h 1 (X ) and h 1 (J X ) denote the Picard motives of X and J X , respectively [1,2], is there an isomorphism ?
h 1 (X ) ≃ h 1 (J X )
(1)
E-mail address: [email protected]. c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights 0019-3577/$ - see front matter ⃝ reserved. doi:10.1016/j.indag.2012.01.005
378
M.D. Kaba / Indagationes Mathematicae 23 (2012) 377–380
In this note, we prove that, as expected, we have such an isomorphism, under the assumption that both the variety X and its Albanese variety J X have dimension at least 2. In fact, this isomorphism is provided by the isomorphism of the Picard varieties of X and J X . 2. Notation and preliminaries The Picard motive was first introduced by Murre in [1]. Later, Scholl gave an excellent exposition of the theory in [2]. Here, we will follow Scholl’s notation with minor exceptions. We write PX for the Picard variety of X , and J X (or A) for its Albanese variety (i.e., the dual of PX ). We denote by d X the dimension of X and by d A the dimension J X . They are both assumed to be greater than 1. For a morphism of varieties φ : X → Y , we denote by Jφ (respectively, Pφ ) the induced morphism J X → JY (respectively, PY → PX ); [Γφ ] denotes the (rational) equivalence class of the graph of φ, and [Γφ ]t denotes the class of the transpose of its graph. If we let σ : X → J X denote the Albanese map (after fixing a point x0 ∈ X and σ (x0 ) = e, the identity ∼ element with respect to the group operation on J X ), we have Jσ = id J X , and Pσ : PJ X → PX is an isomorphism. For the sake of completeness, we include some results of Scholl here. Theorem 1 ([2, Theorem 3.9]). Let X and Y be smooth, projective varieties (defined over some field k) of dimensions d X and dY , respectively. 1. Then there is an isomorphism ∼
Hom Ab (J X , PY ) ⊗ Q −→ p1
A1 (X × Y ) , p1∗ A1 (X ) + p2∗ A1 (Y )
p2
where X ←− X × Y −→ Y are projections and A1 (·) denotes the Chow group of codimension 1 cycles with rational coefficients. 2. Let ζ ∈ Ad X (X ) and η ∈ AdY (Y ) be fixed zero cycles of positive degree. Then there is an isomorphism Ω
Hom Ab (J X , PY ) ⊗ Q → {c ∈ A1 (X × Y )|c ◦ ζ∗ = 0 = η∗ ◦ c}. (For definitions of ζ∗ and η∗ , see [2, p.169].) Proposition 1 ([2, Proposition 3.10]). Let X, Y, X ′ and Y ′ be smooth, projective varieties and φ : X ′ → X, ψ : Y ′ → Y and µ : J X → PY be morphisms. Then Ω (Pψ ◦ µ) = [Γψ ]t ◦ Ω (µ) and
Ω (µ ◦ Jφ ) = Ω (µ) ◦ [Γφ ],
where Ω is the isomorphism in the Theorem 1. Let C X and C A be one-dimensional linear sections of X and J X , respectively. Then ξ X := [Γi X ] ◦ [Γi X ]t
and ξ A := [Γi A ] ◦ [Γi A ]t ,
where i X : C X ↩→ X and i A : C A ↩→ J X denote the embedding maps. We have the induced isogenies Pi X
Ji X
α X : PX → PC X = JC X → J X Pi A
Ji A
α A : PJ X → PC A = JC A → J J X = J X .
M.D. Kaba / Indagationes Mathematicae 23 (2012) 377–380
379
We will denote the inverse isogenies by β X and β A , respectively. By taking a suitable common multiple if necessary, we can write α X ◦ β X = n · id J X = α A ◦ β A . Then, the correspondences p1X :=
1 Ω (β X ) ◦ ξ X , n
X p2d := ( p1X )t , X −1
1 X π1X := p1X ◦ 1 − p2d X −1 2
are projectors, and the Picard motive is given by h 1 (X ) := (X, π1X ). In fact, π1X = p1X when d X ≥ 3 [2]. 3. The isomorphism We would like to prove that h 1 (X ) and h 1 (J X ) are isomorphic in M. In fact, it is enough to prove the isomorphism of (X, p1X ) and (J X , p1A ), as π1X ◦ p1X : (X, p1X ) → h 1 (X ) and p1X ◦ π1X : h 1 (X ) → (X, p1X ) are mutually inverse isomorphisms (of course, the same holds for p1A and π1A ). We define 1 A p ◦ Ω (β A ) ◦ [Γσ ] ◦ ξ X ◦ p1X ∈ HomM ((X, p1X ), (J X , p1A )) n 1 1 θ2 := p1X ◦ Ω (β X ) ◦ ξ A ◦ p1A ∈ HomM ((J X , p1A ), (X, p1X )). n
θ1 :=
Note that, in the definition of θ2 , we interpret β X as a morphism J J X → PX , namely β X = β X ◦ Jσ−1 but Jσ = id J X . We will show that θ1 and θ2 are mutually inverse isomorphisms. θ2 ◦ θ1 = =
= = = =
1 X p ◦ Ω (β X ) ◦ ξ A ◦ p1A ◦ Ω (β A ) ◦ [Γσ ] ◦ ξ X ◦ p1X n2 1 1 X p ◦ Ω (β X ) ◦ [Γi A ] ◦ [Γi A ]t ◦ Ω (β A ) ◦ [Γi A ] ◦ [Γi A ]t n3 1 ◦ Ω (β A ) ◦ [Γσ ] ◦ ξ X ◦ p1X 1 X p ◦ Ω (β X ◦ Ji A ) ◦ Ω (Pi A ◦ β A ◦ Ji A ) ◦ Ω (Pi A ◦ β A ◦ Jσ ) ◦ ξ X ◦ p1X n3 1 1 X p ◦ Ω (β X ◦ Ji A ◦ Pi A ◦ β A ◦ Ji A ◦ Pi A ◦ β A ◦ id J X ) ◦ ξ X ◦ p1X n3 1 1 X p ◦ Ω (β X ◦ α A ◦ β A ◦ α A ◦ β A ) ◦ ξ X ◦ p1X n3 1 p1X ◦ p1X ◦ p1X = p1X .
For the converse, we have 1 A p ◦ Ω (β A ) ◦ [Γσ ] ◦ ξ X ◦ p1X ◦ Ω (β X ) ◦ ξ A ◦ p1A n2 1 1 = 3 p1A ◦ Ω (β A ) ◦ [Γσ ] ◦ [Γi X ] ◦ [Γi X ]t n
θ1 ◦ θ2 =
380
M.D. Kaba / Indagationes Mathematicae 23 (2012) 377–380
◦ Ω (β X ) ◦ [Γi X ] ◦ [Γi X ]t ◦ Ω (β X ) ◦ ξ A ◦ p1A 1 = 3 p1A ◦ Ω (β A ◦ Jσ ◦ Ji X ) ◦ Ω (Pi X ◦ β X ◦ Ji X ) ◦ Ω (Pi X ◦ β X ) ◦ ξ A ◦ p1A n 1 = 3 p1A ◦ Ω (β A ◦ id J X ◦ α X ◦ β X ◦ α X ◦ β X ) ◦ ξ A ◦ p1A n = p1A ◦ p1A ◦ p1A = p1A . Hence, the isomorphism (1) holds. Acknowledgments I started working on this problem during the Ph.D. program I was enrolled in at Middle East Technical University (Ankara), and later continued working on it during my visit to University ¨ of Regensburg. I would like to thank my supervisor Prof. Dr. Hurs¸it Onsiper and Prof. Dr. Uwe Jannsen for their valuable conversations. I am also grateful to the anonymous referee whose remarks on an earlier version of the paper contributed very much. References [1] J.P. Murre, On the motive of an algebraic surface, J. Reine Angew. Math. 409 (1990) 190–204. [2] A.J. Scholl, Classical motives, in: Motives I, in: Proceedings of the Mathematics Symposia, vol. 55, American Mathematical Society, Providence, RI, 1994, pp. 163–187.
Indagationes Mathematicae 23 (2012) 377–380 www.elsevier.com/locate/indag
A note on the Picard motive of a variety Mustafa Devrim Kaba Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Nijenborgh 9, 9747 AG, Groningen, The Netherlands Received 9 May 2010; received in revised form 27 January 2012; accepted 28 January 2012
Communicated by H.W. Broer
Abstract We prove that the Picard motive of a smooth projective variety and the Picard motive of its Albanese variety are isomorphic, under the assumption that both the variety and its Albanese variety have dimension at least 2. c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Chow motives; Picard motive; Albanese variety
1. Introduction For various Weil cohomology theories (H ∗ (·), e.g., singular cohomology, e´ tale cohomology, etc.), we have the well-known isomorphism H 1 (X ) ≃ H 1 (J X ), where X is a smooth projective variety and J X is its Albanese variety, both defined over some field k. Therefore, it is natural to ask if a similar isomorphism holds in the category of pure Chow motives (M). More precisely, if we let h 1 (X ) and h 1 (J X ) denote the Picard motives of X and J X , respectively [1,2], is there an isomorphism ?
h 1 (X ) ≃ h 1 (J X )
(1)
E-mail address: [email protected]. c 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights 0019-3577/$ - see front matter ⃝ reserved. doi:10.1016/j.indag.2012.01.005
378
M.D. Kaba / Indagationes Mathematicae 23 (2012) 377–380
In this note, we prove that, as expected, we have such an isomorphism, under the assumption that both the variety X and its Albanese variety J X have dimension at least 2. In fact, this isomorphism is provided by the isomorphism of the Picard varieties of X and J X . 2. Notation and preliminaries The Picard motive was first introduced by Murre in [1]. Later, Scholl gave an excellent exposition of the theory in [2]. Here, we will follow Scholl’s notation with minor exceptions. We write PX for the Picard variety of X , and J X (or A) for its Albanese variety (i.e., the dual of PX ). We denote by d X the dimension of X and by d A the dimension J X . They are both assumed to be greater than 1. For a morphism of varieties φ : X → Y , we denote by Jφ (respectively, Pφ ) the induced morphism J X → JY (respectively, PY → PX ); [Γφ ] denotes the (rational) equivalence class of the graph of φ, and [Γφ ]t denotes the class of the transpose of its graph. If we let σ : X → J X denote the Albanese map (after fixing a point x0 ∈ X and σ (x0 ) = e, the identity ∼ element with respect to the group operation on J X ), we have Jσ = id J X , and Pσ : PJ X → PX is an isomorphism. For the sake of completeness, we include some results of Scholl here. Theorem 1 ([2, Theorem 3.9]). Let X and Y be smooth, projective varieties (defined over some field k) of dimensions d X and dY , respectively. 1. Then there is an isomorphism ∼
Hom Ab (J X , PY ) ⊗ Q −→ p1
A1 (X × Y ) , p1∗ A1 (X ) + p2∗ A1 (Y )
p2
where X ←− X × Y −→ Y are projections and A1 (·) denotes the Chow group of codimension 1 cycles with rational coefficients. 2. Let ζ ∈ Ad X (X ) and η ∈ AdY (Y ) be fixed zero cycles of positive degree. Then there is an isomorphism Ω
Hom Ab (J X , PY ) ⊗ Q → {c ∈ A1 (X × Y )|c ◦ ζ∗ = 0 = η∗ ◦ c}. (For definitions of ζ∗ and η∗ , see [2, p.169].) Proposition 1 ([2, Proposition 3.10]). Let X, Y, X ′ and Y ′ be smooth, projective varieties and φ : X ′ → X, ψ : Y ′ → Y and µ : J X → PY be morphisms. Then Ω (Pψ ◦ µ) = [Γψ ]t ◦ Ω (µ) and
Ω (µ ◦ Jφ ) = Ω (µ) ◦ [Γφ ],
where Ω is the isomorphism in the Theorem 1. Let C X and C A be one-dimensional linear sections of X and J X , respectively. Then ξ X := [Γi X ] ◦ [Γi X ]t
and ξ A := [Γi A ] ◦ [Γi A ]t ,
where i X : C X ↩→ X and i A : C A ↩→ J X denote the embedding maps. We have the induced isogenies Pi X
Ji X
α X : PX → PC X = JC X → J X Pi A
Ji A
α A : PJ X → PC A = JC A → J J X = J X .
M.D. Kaba / Indagationes Mathematicae 23 (2012) 377–380
379
We will denote the inverse isogenies by β X and β A , respectively. By taking a suitable common multiple if necessary, we can write α X ◦ β X = n · id J X = α A ◦ β A . Then, the correspondences p1X :=
1 Ω (β X ) ◦ ξ X , n
X p2d := ( p1X )t , X −1
1 X π1X := p1X ◦ 1 − p2d X −1 2
are projectors, and the Picard motive is given by h 1 (X ) := (X, π1X ). In fact, π1X = p1X when d X ≥ 3 [2]. 3. The isomorphism We would like to prove that h 1 (X ) and h 1 (J X ) are isomorphic in M. In fact, it is enough to prove the isomorphism of (X, p1X ) and (J X , p1A ), as π1X ◦ p1X : (X, p1X ) → h 1 (X ) and p1X ◦ π1X : h 1 (X ) → (X, p1X ) are mutually inverse isomorphisms (of course, the same holds for p1A and π1A ). We define 1 A p ◦ Ω (β A ) ◦ [Γσ ] ◦ ξ X ◦ p1X ∈ HomM ((X, p1X ), (J X , p1A )) n 1 1 θ2 := p1X ◦ Ω (β X ) ◦ ξ A ◦ p1A ∈ HomM ((J X , p1A ), (X, p1X )). n
θ1 :=
Note that, in the definition of θ2 , we interpret β X as a morphism J J X → PX , namely β X = β X ◦ Jσ−1 but Jσ = id J X . We will show that θ1 and θ2 are mutually inverse isomorphisms. θ2 ◦ θ1 = =
= = = =
1 X p ◦ Ω (β X ) ◦ ξ A ◦ p1A ◦ Ω (β A ) ◦ [Γσ ] ◦ ξ X ◦ p1X n2 1 1 X p ◦ Ω (β X ) ◦ [Γi A ] ◦ [Γi A ]t ◦ Ω (β A ) ◦ [Γi A ] ◦ [Γi A ]t n3 1 ◦ Ω (β A ) ◦ [Γσ ] ◦ ξ X ◦ p1X 1 X p ◦ Ω (β X ◦ Ji A ) ◦ Ω (Pi A ◦ β A ◦ Ji A ) ◦ Ω (Pi A ◦ β A ◦ Jσ ) ◦ ξ X ◦ p1X n3 1 1 X p ◦ Ω (β X ◦ Ji A ◦ Pi A ◦ β A ◦ Ji A ◦ Pi A ◦ β A ◦ id J X ) ◦ ξ X ◦ p1X n3 1 1 X p ◦ Ω (β X ◦ α A ◦ β A ◦ α A ◦ β A ) ◦ ξ X ◦ p1X n3 1 p1X ◦ p1X ◦ p1X = p1X .
For the converse, we have 1 A p ◦ Ω (β A ) ◦ [Γσ ] ◦ ξ X ◦ p1X ◦ Ω (β X ) ◦ ξ A ◦ p1A n2 1 1 = 3 p1A ◦ Ω (β A ) ◦ [Γσ ] ◦ [Γi X ] ◦ [Γi X ]t n
θ1 ◦ θ2 =
380
M.D. Kaba / Indagationes Mathematicae 23 (2012) 377–380
◦ Ω (β X ) ◦ [Γi X ] ◦ [Γi X ]t ◦ Ω (β X ) ◦ ξ A ◦ p1A 1 = 3 p1A ◦ Ω (β A ◦ Jσ ◦ Ji X ) ◦ Ω (Pi X ◦ β X ◦ Ji X ) ◦ Ω (Pi X ◦ β X ) ◦ ξ A ◦ p1A n 1 = 3 p1A ◦ Ω (β A ◦ id J X ◦ α X ◦ β X ◦ α X ◦ β X ) ◦ ξ A ◦ p1A n = p1A ◦ p1A ◦ p1A = p1A . Hence, the isomorphism (1) holds. Acknowledgments I started working on this problem during the Ph.D. program I was enrolled in at Middle East Technical University (Ankara), and later continued working on it during my visit to University ¨ of Regensburg. I would like to thank my supervisor Prof. Dr. Hurs¸it Onsiper and Prof. Dr. Uwe Jannsen for their valuable conversations. I am also grateful to the anonymous referee whose remarks on an earlier version of the paper contributed very much. References [1] J.P. Murre, On the motive of an algebraic surface, J. Reine Angew. Math. 409 (1990) 190–204. [2] A.J. Scholl, Classical motives, in: Motives I, in: Proceedings of the Mathematics Symposia, vol. 55, American Mathematical Society, Providence, RI, 1994, pp. 163–187.