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," $X$ 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 $F^1H^n(X,\mathbb{C})=H^{1,n-1}(X)\oplus\cdots\oplus H^{n,0}(X)$ 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 $2^n-N$, 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.


This definitely looks like an error in Grothendieck's paper! I think I have figured out what he meant. To simplify notation, I will concentrate on the case where Grothendieck builds his example: $H^3$ of a product of $3$ elliptic curves. I prefer to think in homology for this purpose, so I'll be talking about $H_3(E_1 \times E_2 \times E_3)$.

Put $X = E_1 \times E_2 \times E_3$ where $E_i = \mathbb{C}/(\mathbb{Z}+\mathbb{Z} \tau_i)$. I'll write $z_i$ for the coordinate on the universal cover of $E_i$, so $dz_i$ is the nonvanishing holomorphic $(1,0)$-form on $E_i$.

Let $V$ be the subspace of $H_3(X, \mathbb{Q})$ spanned by the images of all $H_3(Y, \mathbb{Q})$, as $Y$ ranges over $2$-dimensional subvarieties of $X$. Hodge's "general conjecture" is an attempted description of $V$. (The famous Hodge conjecture deals with $H_{2 \dim Y}(Y)$, so we are talking about the span of the fundamental classes; this less famous one allows the subscript on homology to be independent of the dimension of $Y$.)

Specifically, Hodge conjectures that $\alpha \in H_3(X, \mathbb{Q})$ lies in $V$ if and only if $\int_{\alpha} \omega=0$ for any $\omega \in H^{(3,0)}(X)$. In fact, $H^{(3,0)}(X)$ is one dimensional, spanned by $d z_1 \wedge d z_2 \wedge dz_3$, so this criterion says that $\int_{\alpha} d z_1 \wedge d z_2 \wedge dz_3=0$. Let's denote the space of $\alpha$ obeying this condition by $V'$. It is easy to see that $V \subseteq V'$, since $ d z_1 \wedge d z_2 \wedge dz_3|_Y=0$ for any algebraic surface $Y$.

Grothendieck observes that, for "trivial reasons", $V$ is even dimensional. However, he shows that $V'$ need not be.

By Kunneth, we have $$H_3(X, \mathbb{Q}) = \bigoplus_{i_1+i_2+i_3=3} H_{i_1}(E_1, \mathbb{Q}) \otimes H_{i_2}(E_2, \mathbb{Q}) \otimes H_{i_3}(E_3, \mathbb{Q}).$$ Every summand other than $H_1 \otimes H_1 \otimes H_1$ is easily seen to be in $V$. So it is natural to consider the inclusion $V \cap (H_1\otimes H_1 \otimes H_1) \subseteq V' \cap (H_1\otimes H_1 \otimes H_1)$, which will be equality if and only if $V \subseteq V'$ is equality. I believe that Grothendieck, without bothering to say so, has switched to working with these subspaces in his dimension computation. Each of the other summands is two-dimensional, so this doesn't effect the parity of the dimension.

So, let's compute the dimension of $V' \cap (H_1\otimes H_1 \otimes H_1)$. Let $\gamma^0_i$ be the $1$-cycle $\mathbb{R}/\mathbb{Z}$ in $E_i$, and let $\gamma^1_i$ be the $1$-cycle $\mathbb{R} \tau_i/\mathbb{Z} \tau_i$. So $\gamma^0_i$, $\gamma^1_i$ is a basis of $H_1(E, \mathbb{Q})$. So a basis for $H_1 \otimes H_1 \otimes H_1$ is the $8$ products $\gamma_1^{j_1} \times \gamma_2^{j_2} \times \gamma_3^{j_3}$, where each $j_i$ is $0$ or $1$. We have $\int_{\gamma_i^j} dz_i = \tau_i^j$, so $$\int_{\gamma_1^{j_1} \times \gamma_2^{j_2} \times \gamma_3^{j_3}} d z_1 \wedge d z_2 \wedge dz_3 = \tau_1^{j_1} \tau_2^{j_2} \tau_3^{j_3}.$$ Putting $$\alpha = \sum_{j_1, j_2, j_3 =0}^1 c(j_1, j_2, j_3)\ [\gamma_1^{j_1} \times \gamma_2^{j_2} \times \gamma_3^{j_3}] \in H_3(X, \mathbb{Q}),$$ we see that $\alpha \in V'$ if and only if $$\sum_{j_1, j_2, j_3 =0}^1 c(j_1, j_2, j_3) \ \tau_1^{j_1} \tau_2^{j_2} \tau_3^{j_3}=0.$$ Thus, $V' \cap (H_1\otimes H_1 \otimes H_1)$ is identified with the $\mathbb{Q}$-vector space $$\left\{ c \in \mathbb{Q}^{\{0,1\}^3} : \sum_{j_1, j_2, j_3 =0}^1 c(j_1, j_2, j_3) \ \tau_1^{j_1} \tau_2^{j_2} \tau_3^{j_3}=0 \right\}.$$ We see that the dimension of $V' \cap (H_1\otimes H_1 \otimes H_1)$ is $2^3-N$ where $N$ is the dimension of the $\mathbb{Q}$-vector space spanned by the monomials $\tau_1^{j_1} \tau_2^{j_2} \tau_3^{j_3}$.

As Grothendieck points out, if $\tau_1=\tau_2=\tau_3$ is a cubic irrational, these monomials span a $3$-dimensional vector space. As you say, for generic $\tau_i$, the monomomials span an $8$ dimensional vector space, so $V' \cap (H_1\otimes H_1 \otimes H_1)=0$. This reflects that the obvious classes, coming from $H_3(E_i \times E_j)$ with $1 \leq i < j \leq 3$, are all in the other summands of $H_3(X)$.


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