Some remarks on the Hodge conjecture for abelian varieties
Let \(X\)be a smooth complex projective variety of dimension \(g\). The general Hodge conjecture \(GHC(X,m,p)\) asserts that for every \(\mathbb{Q}\)-Hodge substructure \(V\) of \(H^m(X,\mathbb{Q})\) with level \(\leq m-2p\), there exists a subvariety \(Z\subset X\) of pure codimension \(p\) such that \(V\subset \ker\{H^m(X,\mathbb{Q})\rightarrow H^m(X-Z,\mathbb{Q})\}\). Let \(Y\) be a smooth ample divisor in \(X\), and let \(K(Y,\mathbb{Q})\) denote the primitive part of the cohomology of \(Y\), namely the kernel of the Gysin map \(H^{g-1}(Y,\mathbb{Q})\rightarrow H^{g+1}(Y,\mathbb{Q})\). We say that \(GHC'(Y,p)\) holds if every Hodge substructure of level \(\leq g-1-2p\) of \(K(Y,\mathbb{Q})\) is supported on a pure codimension \(p\) subvariety of \(Y\). In this paper, the author shows among other things that if \(m\leq g-2\), then \(GHC(Y,m,p)\) holds if and only if \(GHC(X,m,p)\) holds. Moreover she proves that \(GHC(Y,g-1,p)\) holds if and only if both \(GHC(X,g-1,p)\) and \(GHC'(Y,p)\) hold. An application to the study of abelian varieties of Weil type is also given.
scientific article; zbMATH DE number 5790120