Comments on Sampson's approach toward Hodge conjecture on Abelian varieties

Let $A$ be an Abelian variety of dimension $n$. For $0<p<2n$ an odd integer, Sampson constructed a surjective homomorphism $\pi :J^p(A)\rightarrow A$, where $J^p(A)$ is the higher Weil Jacobian variety of $A$. Let $\widehat{\omega}$ be a fixed form in $H^{1,1}(J^p(A),\mathbb{Q})$, and $N=\dim (J^p(A))$. He observes that if the map $\pi _*(\widehat{\omega }^{N-p-1}\wedge .): H^{1,1}(J^p(A),\mathbb{Q})\rightarrow H^{n-p,n-p}(A,\mathbb{Q})$ is injective, then the Hodge conjecture is true for $A$ in bidegree $(p,p)$. In this paper, we give some clarification of the approach and show that the map above is {not injective} except some special cases where the Hodge conjecture is already known. We propose a modified approach.