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 be projective variety of (complex) dimension and let an inclusion of a subvariety of codimension . We have the induced map on homology
After taking Poincare duality on both sides, we get
This is a morphism of Hodge structures, and we should do a Tate twist to have the same weights:
Definition 1 A Hodge structure of weight has coniveau if , but when or .
So, we can say that the image of above is a Hodge substructure of coniveau or higher. A different description of this image is as the kernel of
The Generalized Hodge Conjecture states that if is a coniveau Hodge substructure, then there exists a subvariety of codimension at lest such that is contained in the kernel of the map above.
The original conjecture of Hodge (refuted by Grothendieck in the article under discussion) was that comes from subvarieties 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 be elliptic curves with
where and form an integral basis for cohomology. Take a product of such curves and consider the group . We have that
Moreover, the filtration element is the symplectic orthogonal of . We thus have
To find , after applying Poincare duality, we see it is the same as
But is generated by the products of and , so
where the product ranges over all possible -tuples of distinct ‘s. In particular, this rank (denoted ) can be odd if and is a cubic irrational number. Since , this rank is odd and so the group cannot carry an odd weight Hodge structure.
Going back to the Hodge conjecture, note that on the coniveau generalized conjecture is the same as the usual one. Let us show (following Grothendieck) that on the coniveau Hodge conjecture is implied by the usual Hodge conjecture.
Given a coniveau Hodge substructre, we see that
After shift (and using the polarization), this gives a polarized Hodge structure of weight . This, in turn, is the same as a polarized abelian variety (up to isogeny).
Take a curve which is a complete intersection. The Lefschetz hyperplane theorem implies we have an injection
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
This gives a rational cycle of degree . Assuming the usual Hodge conjecture, this should be represented by some cycle .
The image of to will then gives the desired variety which contains the Hodge substructre . Some care is needed about desingularizing and taking the components which project with the correct dimension.