Abstract
For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.
Highlights
Let H be a pure integral Hodge structure of weight-(−1)
Walker has shown that the Abel–Jacobi map on algebraically trivial cycle classes lifts to the Walker intermediate Jacobian: Theorem A (Walker, [23]) Let X be a complex projective manifold
Note that since for a smooth complex projective variety X one has that A p(X ) is a divisible group (e.g., [9, Lem. 7.10]), there is at most one homomorphism ψWp : A p(X ) → JW2p−1(X ) such that α ◦ ψWp = ψ p; i.e., there is at most one lifting of the Abel–Jacobi map to the Walker intermediate Jacobian
Summary
We start with the following elementary fact, which will be used recurringly throughout this note. Assume D is divisible and that ker α is finite. There exists at most one homomorphism f : D → G such that α ◦ f = f , i.e., such that the following diagram commutes: G f α. Note that since for a smooth complex projective variety X one has that A p(X ) is a divisible group (e.g., [9, Lem. 7.10]), there is at most one homomorphism ψWp : A p(X ) → JW2p−1(X ) such that α ◦ ψWp = ψ p; i.e., there is at most one lifting of the Abel–Jacobi map to the Walker intermediate Jacobian
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