Distinguished Models of Intermediate Jacobians

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.

Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for Abelian varieties

“…15]. 4 For type III Murty finds that L λ ⊗ R C is a special orthogonal group ; this is because, contrary to the convention we adopted here, he considers the connected component of the identity of the Lefschetz group. Finally, we observe that, since D ⊗ Q C ∼ = End Hdg(A)C (H 1 (A, C)), the action of D ⊗ Q C on V (A) C = H 1 (A, C) commutes with the Hodge decomposition.…”

Normal Functions for Algebraically Trivial Cycles Are Algebraic for Arithmetic Reasons

“…In the case where B is quasiprojective, X = B × Y for some smooth projective complex manifold Y, and f : X → B is the first projection, part (1) of the theorem is elementary by embedding B in a smooth complex projective manifold B, extending the cycle class Z to a cycle class Z on B × Y, and obtaining a normal function ν Z : B → B × J 2n+1 a (Y) that is a holomorphic map between complex projective manifolds. From our work in [ACMV18], one can then easily deduce…”

“…The starting point of our proof consists in showing that, for a smooth projective variety X defined over a subfield K ⊆ C, the kernel of the Abel-Jacobi map restricted to algebraically trivial cycles defined over K is independent of the choice of field embedding K ⊆ C. This is embodied in Corollary 3.3; in fact, a stronger result is proved in Proposition 3.1 where it is shown that the distinguished model of [ACMV18] does not depend on a choice of field embedding. The proof uses in an essential way the fact proven in [ACMV19] that algebraically trivial cycles defined over K are parameterized by abelian varieties, and builds on our previous work [ACMV18]. Consequently, the relevant material of [ACMV18] is reviewed in Section 2.…”

“…The proof uses in an essential way the fact proven in [ACMV19] that algebraically trivial cycles defined over K are parameterized by abelian varieties, and builds on our previous work [ACMV18]. Consequently, the relevant material of [ACMV18] is reviewed in Section 2. An important consequence of Proposition 3.1 is that an algebraically F-motivated normal function vanishes at a very general point if and only if it vanishes on a Zariski open subset and is therefore identically zero; see Example 3.6 and Remarks 3.7 and 3.8.…”

Derived equivalent threefolds, algebraic representatives, and the coniveau filtration

“…To a smooth projective variety X over a subfield of C one may associate the total image of the Abel-Jacobi map restricted to algebraically trivial cycles, J a (X C ), an abelian variety defined over K ( §2, [ACV16]) that sits inside the total intermediate Jacobian. We show (Proposition 2.1) that if X and Y are derived equivalent smooth projective varieties of arbitrary dimension over a subfield of C, then J a (X C ) and J a (Y C ) are isogenous over K ; over C this provides a short proof of a special case of a result of [BT16], which also considers semi-orthogonal decompositions.…”