Let $X=E_1\times\cdots\times E_n$, where $E_i$ is the elliptic curve $E_i=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\alpha_i)$. In Grothendieck's ["The Hodge Conjecture is False for Trivial Reasons,"][1] $X$ is stated to fail the original formulation of the Generalized Hodge Conjecture, for appropriate choices of $\alpha_i$. Namely, there exist classes in $F^1H^n(X,\mathbb{C})\cap H^n(X,\mathbb{Q})$ that are not generated by images of Gysin maps $H^{n-2q}(Y,\mathbb{Q})\to H^{n}(X,\mathbb{Q})$, where $Y$ is a resolution of a codimension $q\ge1$ subscheme of $X$. (Here $F^1H^n(X,\mathbb{C})=H^{1,n-1}(X)\oplus\cdots H^{n,0}(X)$ is the first term in the Hodge filtration for $X$.)

He states without proof that the dimension of the subspace of $H^n(X,\mathbb{Q})$ generated by images of Gysin maps is $2^i-N$, where $N$ is the dimension of the $\mathbb{Q}$-vector space generated by distinct products of $\alpha_i$. I am confused by this, because it seems that when the $\alpha_i$ are generic, this number is 0. In particular, there are no non-zero Hodge classes in $H^2(E_1\times E_2,\mathbb{Q})$, but don't $E_1\times 0$ and $0\times E_2$ define nonzero Hodge classes?

Would appreciate it if someone could clear up my confusion and also provide a hint for how to prove the claim in the above paragraph.

  [1]: https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/HodgeConj.pdf