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.
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