Back to blogging for a bit, though likely infrequently. Doing a new series that might count as AG from the beginning, so I’ll put it up there once I’ve got a couple done. We’re going to start doing some Hodge Theory.
We start with a finite rank lattice , and we’re going to be working with its complexification .
Def: A Hodge Filtration of weight is a descending filtration on such that for all we have .
Def: A Hodge Decomposition of weight is a decomposition such that .
As we’re living in the complex numbers, and have a distinguished real subspace, we’re going to be conjugating a lot, and that’s exactly what the overlines are. If the notation turns out to be bad, I can always change it later.
Here’s the important theorem:
Prop: The above are equivalent.
Pf: Start with a filtration. We define . Then as needed. The decomposition itself is merely some messy linear algebra.
If we start with a decomposition, then taking gives us a filtration, and it is easily verified that the two properties hold.
We’ll call any Hodge filtration or Hodge decomposition a Hodge structure. This is an important concept, and eventually we’ll get to constructing a lot of examples, which will also show why they’re useful, by showing that every smooth variety has Hodge structures on its cohomology.
Now, a few constructions:
Let be a Hodge structure of weight and let be a Hodge structure of weight . Then we can look at . This is naturally a Hodge structure of weight , and in particular sits inside of . Later, this will be connected with the Künneth Isomorphism.
Next, let be a Hodge structure of weight . What can we say about the space ? For one thing, we should get a trivial object with . If we guess that this product should have a Hodge structure of weight , then that tells us that should have a Hodge structure of weight . So what happens to ? Well, the only natural thing is to call it , and this is exactly the correct answer to make everything work out.
With duals and tensor products under our belts, we can give a Hodge structure, of weight , as . With this, we can define morphisms of Hodge structures, which will make Hodge structures into a category. There are two possible choices for morphisms: the most natural one is the elements of . These only give morphisms of Hodge structures of the same weight, though. The other choice is to take all classes in for all . We can relate the two via Tate twists, but I’m not going to go in that direction.
The last construction, a personal favorite, is of the Jacobian complex torus associated to a Hodge structure. Let be a Hodge structure of weight . For this, we use the filtration. We have that . Quotient by , and then further quotient by the image of the lattice . The claim is that this will give us a torus (that there is a complex structure should be clear). The only question is if is a lattice of full rank in . The dimensions are rank match up, so we just need to check that this map is injective. That, in fact, is equivalent to having . We have this! It’s just because for elements of the lattice , they are mapped into to a pair of complex conjugate elements, and so quotienting by either side would give an inclusion. Thus, we have a complex torus, which we’ll denote by . Later, we’ll apply this to the Hodge structure of weight on to construct the intermediate Jacobians of varieties, which carry a lot of (maddeningly mysterious) information about the variety itself.