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<r} or {q<r}.

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.

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