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DISTINGUISHED MODELS of INTERMEDIATE JACOBIANS Let X

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DISTINGUISHED MODELS of INTERMEDIATE JACOBIANS Let X DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS JEFFREY D. ACHTER, SEBASTIAN CASALAINA-MARTIN, AND CHARLES VIAL Abstract. We show that the image of the Abel{Jacobi map admits functorially a model over the field of definition, with the property that the Abel{Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level one. Let X be a smooth projective variety defined over the complex numbers. Given a nonnegative integer n, denote CHn+1(X) the Chow group of codimension-(n+1) cycle classes on X, and denote n+1 n+1 2(n+1) CH (X)hom the kernel of the cycle class map CH (X) ! H (X; Z(n + 1)): In the seminal paper [Gri69], Griffiths defined a complex torus, the intermediate Jacobian, J 2n+1(X) together with an Abel{Jacobi map n+1 2n+1 AJ : CH (X)hom ! J (X): While J 2n+1(X) and the Abel{Jacobi map are transcendental in nature, the image of the Abel{ Jacobi map restricted to An+1(X), the sub-group of CHn+1(X) consisting of algebraically trivial 2n+1 2n+1 cycle classes, is a complex sub-torus Ja (X) of J (X) that is naturally endowed via the Hodge bilinear relations with a polarization, and hence is a complex abelian variety. The first cohomology 2n+1 n 2n+1 group of Ja (X) is naturally identified via the polarization with N H (X; Q(n)) ; i.e., the n-th Tate twist of the n-th step in the geometric coniveau filtration (see (1.1)). If now X is a smooth projective variety defined over a sub-field K ⊆ C, it is natural to ask 2n+1 whether the complex abelian variety Ja (XC) admits a model over K. In this paper, we prove 2n+1 that Ja (XC) admits a unique model over K such that n+1 2n+1 AJ :A (XC) ! Ja (XC) is Aut(C=K)-equivariant, thereby generalizing the well-known cases of the Albanese map dim X 1 0 A (X ) ! AlbX and of the Picard map A (X ) ! Pic , as well as the case of AJ : C C C XC 2 3 A (XC) ! Ja (XC) which was treated in our previous work [ACMV17a]. Theorem A. Suppose X is a smooth projective variety over a field K ⊆ C, and let n be a non- 2n+1 negative integer. Then Ja (XC), the complex abelian variety that is the image of the Abel{ n+1 2n+1 Jacobi map AJ :A (XC) ! J (XC), admits a distinguished model J over K such that the Abel{Jacobi map is Aut(C=K)-equivariant. Moreover, there is an algebraic correspondence dim(J)+n Γ 2 CH (J ×K X) inducing, for every prime number `, a split inclusion of Gal(K)- representations 1 2n+1 (0.1) Γ∗ : H (JK ; Q`) ,! H (XK ; Q`(n)) n 2n+1 with image N H (XK ; Q`(n)). Date: April, 2018. The first author was partially supported by grants from the NSA (H98230-14-1-0161, H98230-15-1-0247 and H98230-16-1-0046). The second author was partially supported by a Simons Foundation Collaboration Grant for Mathematicians (317572) and NSA grant H98230-16-1-0053. The third author was supported by EPSRC Early Career Fellowship EP/K005545/1. 1 By Chow's rigidity theorem (see [Con06, Thm. 3.19]), an abelian variety A=C descends to at most one model defined over K. On the other hand, an abelian variety A=K may descend to more n+1 2n+1 than one model defined over K. Nevertheless, since AJ :A (XC) ! Ja (XC) is surjective, 2n+1 the abelian variety Ja (XC) admits at most one structure of a scheme over K such that AJ is 2n+1 Aut(C=K)-equivariant. This is the sense in which Ja (XC) admits a distinguished model over K. Our proof of TheoremA uses a different strategy than we took in [ACMV17a], and as a result improves on the results of that paper in three ways : 1. In [ACMV17a, Thm. B], only the case n = 1 of TheoremA was treated. An essential step in the proof in [ACMV17a, Thm. B] was a result of Murre [Mur85, Thm. C], relying on the theorem 3 of Merkurjev and Suslin, asserting that Ja (XC) is an algebraic representative, meaning that it is 2 universal among regular homomorphisms from A (XC) (as defined in x3). In general, little is known about when higher intermediate Jacobians are algebraic representatives, or even when algebraic representatives exist. In this paper we completely avoid the use of Murre's result, or indeed the existence of an algebraic representative. Instead, we use a new approach to show that for each n 2n+1 there is a model of Ja (XC) over K which makes the Abel{Jacobi map Aut(C=K)-equivariant. 2n+1 2. The results of [ACMV17a, Thm. A] concerning descent for Ja (XC) for n > 1 only show that 2n+1 the isogeny class of Ja (XC) descends to K, and this is under the further restrictive assumption that the Abel{Jacobi map be surjective (or under some other constraint on the motive of X ; see [ACMV17a, Thm. 2.1]). In contrast, the present TheoremA provides a distinguished model 2n+1 of Ja (XC) over K, without any additional hypothesis. Moreover, we show the assignment in TheoremA is functorial (Proposition 5.1). The new technical input begins with Proposition 2n+1 1.1, which shows that Ja (XC) is dominated, via the induced action of a correspondence defined over K, by the Jacobian of a pointed, geometrically integral, smooth projective curve C defined over K, strengthening [ACMV17a, Prop. 1.3]. The key point is that this strengthening, together with the fact that Bloch's map [Blo79] factors through the Abel{Jacobi map on torsion, makes it possible 2n+1 to show directly that Ja (XC) admits a unique model over K making the Abel{Jacobi map n+1 2n+1 AJ :A (XC) ! Ja (XC) Galois equivariant on torsion. In short, avoiding the use of algebraic representatives, and the motivic techniques employed in [ACMV17a], we obtain a stronger result. We then make a careful analysis of Galois equivariance for regular homomorphisms, strengthening some statements in [ACMV17a], to conclude that the Abel{Jacobi map is Galois equivariant on all points { and not merely on torsion points (Proposition 3.8) ; this is crucial to the proof of TheoremB below. 3. Finally, while a splitting in [ACMV17a, Thm. A] analogous to (0.1) was established by some explicit computations involving correspondences, here we utilize Andr´e'spowerful theory of mo- tivated cycles [And96] in order to establish the more general splitting (0.1). This also provides a proof that the coniveau filtration splits (Corollary 4.4), as well as a short motivic proof that the 2n+1 isogeny class of Ja (XC) descends, without any of the restrictive hypotheses in [ACMV17a]. The structure of the proof of TheoremA is broken into three parts. First we give a proof of TheoremA, up to the statement of the splitting of the inclusion, and where we focus only on the Aut(C=K)-equivariance of the Abel{Jacobi map on torsion (Theorem 2.1). The proof of Theo- n 2n+1 rem 2.1 relies on showing that N H (XK ; Q`(n)) is spanned via the action of a correspondence over K by the first cohomology group of a pointed, geometrically integral curve ; this is proved in Proposition 1.1. Next, in x3, we show that if the Abel{Jacobi map is Aut(C=K)-equivariant on torsion, then it is fully equivariant. This is a consequence of more general results we establish for surjective regular homomorphisms. Finally, the splitting of (0.1) is then proved in Theorem 4.2. Note that when n = 1 the result of [ACMV17a, Thm. A] is more precise in that the splitting of (0.1) is shown to be induced by an algebraic correspondence over K. 2 As a first application of TheoremA, we recover a result of Deligne [Del72] regarding intermediate Jacobians of complete intersections of Hodge level 1 (x6). Another application is to the following question due to Barry Mazur. Given an effective polariz- able weight-1 Q-Hodge structure V , there is a complex abelian variety A (determined up to isogeny) 1 so that H (A; Q) gives a weight-1 Q-Hodge structure isomorphic to V . On the other hand, let K be a field, and let ` be a prime number (not equal to the characteristic of the field). It is not known (even for K = Q) whether given an effective polarizable weight-1 Gal(K)-representation V` over Q`, 1 there is an abelian variety A=K such that H (AK ; Q`) is isomorphic to V`.A phantom abelian va- riety for V` is an abelian variety A=K together with an isomorphism of Gal(K)-representations ∼ 1 = / H (AK ; Q`) V`: Such an abelian variety, if it exists, is determined up to isogeny ; this is called the phantom isogeny class for V`.
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