Grothendieck and the (generalized) Hodge conjecture

Grothendieck famously wrote a paper called Hodge’s general conjecture is false for trivial reasons. In this post, I would like to record, for my own benefit, some of the observations made in it.

Let ${X^n}$ be projective variety of (complex) dimension ${n}$ and let ${\iota:Y^{n-c}{\hookrightarrow} X^n}$ an inclusion of a subvariety of codimension ${c}$. We have the induced map on homology

$\displaystyle \iota_* : H_\bullet Y {\rightarrow} H_\bullet X$

After taking Poincare duality on both sides, we get

$\displaystyle \iota_!:H^\bullet(Y){\rightarrow} H^{\bullet+2c}(X)$

This is a morphism of Hodge structures, and we should do a Tate twist to have the same weights:

$\displaystyle \iota_!:H^\bullet(Y)(-c){\rightarrow} H^{\bullet+2c}(X)$

Definition 1 A Hodge structure ${H_{\mathbb Z}}$ of weight ${w}$ has coniveau ${r}$ if ${H^{w-r,r}\neq 0}$, but ${H^{p,q}=0}$ when ${p or ${q.

So, we can say that the image of ${\iota_!}$ above is a Hodge substructure of coniveau ${c}$ or higher. A different description of this image is as the kernel of

$\displaystyle j^*:H^\bullet(X) {\rightarrow} H^\bullet(X\setminus Y)$

The Generalized Hodge Conjecture states that if ${L\subset H^kX}$ is a coniveau ${c}$ Hodge substructure, then there exists a subvariety ${Y}$ of codimension at lest ${c}$ such that ${L}$ is contained in the kernel of the map ${j^*}$ above.

The original conjecture of Hodge (refuted by Grothendieck in the article under discussion) was that ${F^c\cap H^k_{\mathbb Q} X}$ comes from subvarieties ${Y}$ as above. Grothendieck’s observation was that these groups need not form a Hodge substructre, so the conjecture cannot be true in the original form.

Example 1 Let ${E_i}$ be elliptic curves with

$\displaystyle H^1_{\mathbb C}(E_i) = dz_i\oplus \overline{dz_i}$

where ${dz_i = e_i + \tau_i f_i}$ and ${e_i,f_i}$ form an integral basis for cohomology. Take a product of ${n}$ such curves and consider the group ${H^n(E_1\times \cdots \times E_n)}$. We have that

$\displaystyle H^{n,0} = \left< \omega := dz_1\wedge\cdots \wedge dz_n\right>$

Moreover, the filtration element ${F^1H^n}$ is the symplectic orthogonal of ${H^{n,0}}$. We thus have

$\displaystyle F^1 = \ker \left((-)\cup \omega: H^n {\rightarrow} H^{2n}\right)$

To find ${F^1\cap H^n_{\mathbb Q}}$, after applying Poincare duality, we see it is the same as

$\displaystyle \ker \left(\int_ \omega :H_{n,{\mathbb Q}}{\rightarrow} {\mathbb C}\right)$

But ${H_{n,{\mathbb Q}}}$ is generated by the products of ${e_i}$ and ${f_j}$, so

$\displaystyle \dim_{\mathbb Q} {\mathrm{im}}\left(\int \omega\right) = \dim_{\mathbb Q} \left(\{\tau_{i_1}\cdots\tau_{i_k}\}\right)$

where the product ranges over all possible ${k}$-tuples of distinct ${\tau}$‘s. In particular, this rank (denoted ${N}$) can be odd if ${n=3}$ and ${\tau_1=\tau_2=\tau_3}$ is a cubic irrational number. Since ${\dim_{\mathbb Q} F^1\cap H^n_{\mathbb Q}= 2^i - N}$, this rank is odd and so the group cannot carry an odd weight Hodge structure.

Going back to the Hodge conjecture, note that on ${H^{2c}}$ the coniveau ${c}$ generalized conjecture is the same as the usual one. Let us show (following Grothendieck) that on ${H^{2c+1}}$ the coniveau ${c}$ Hodge conjecture is implied by the usual Hodge conjecture.

Given ${L\subset H^{2c+1}X}$ a coniveau ${c}$ Hodge substructre, we see that

$\displaystyle L\cong L^{c+1,c}\oplus L^{c,c+1}$

After shift (and using the polarization), this gives a polarized Hodge structure of weight ${1}$. This, in turn, is the same as a polarized abelian variety ${A}$ (up to isogeny).

Take a curve ${C\subset A}$ which is a complete intersection. The Lefschetz hyperplane theorem implies we have an injection

$\displaystyle H^1 A {\hookrightarrow} H^1 C$

We have thus realized any abelian variety as a direct factor of the Jacobian of some curve (up to isogeny). Using the polarization to take complements, we thus have a morphism of Hodge structures

$\displaystyle H^1 C {\rightarrow} H^{2c+1} X$

This gives a rational cycle ${[M]\in H^{2c+2}(C\times X)}$ of degree ${(c+1,c+1)}$. Assuming the usual Hodge conjecture, this should be represented by some cycle ${Z\subset C\times X}$.

The image of ${Z}$ to ${X}$ will then gives the desired variety ${Y}$ which contains the Hodge substructre ${L}$. Some care is needed about desingularizing ${Z}$ and taking the components which project with the correct dimension.