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

TL;DR

This paper critically examines Sampson's approach to the Hodge conjecture on Abelian varieties, showing the key map's non-injectivity generally and proposing a revised method for progress.

Contribution

The paper clarifies Sampson's approach and demonstrates the non-injectivity of the critical map, suggesting a modified strategy for tackling the Hodge conjecture.

Findings

The map in Sampson's approach is not injective in general.

Injectivity holds only in special cases where the Hodge conjecture is already proven.

A new approach is proposed to address the limitations of the original method.

Abstract

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.