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 H_\mathbb{Z}, and we’re going to be working with its complexification H_\mathbb{C}.

Def: A Hodge Filtration of weight k is a descending filtration on H_\mathbb{C} such that for all p we have F^p\oplus\overline{F^{n-p+1}}=H_\mathbb{C}.

Def: A Hodge Decomposition of weight k is a decomposition H_\mathbb{C}=\oplus_{p+q=k}H^{p,q} such that H^{p,q}=\overline{H^{q,p}}.

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 H^{p,q}=F^p\cap \overline{F^q}.  Then \overline{H^{p,q}}=\overline{F^p}\cap F^q=H^{q,p} as needed.  The decomposition itself is merely some messy linear algebra.

If we start with a decomposition, then taking F^p=\oplus_{p'\geq p} H^{p',q} gives us a filtration, and it is easily verified that the two properties hold. \Box

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 H be a Hodge structure of weight h and let K be a Hodge structure of weight k.  Then we can look at H\otimes K.  This is naturally a Hodge structure of weight h+k, and in particular H^{p,q}\otimes K^{a,b} sits inside of (H\otimes K)^{p+a,q+b}.  Later, this will be connected with the Künneth Isomorphism.

Next, let H be a Hodge structure of weight h.  What can we say about the space H^\vee? For one thing, we should get a trivial object with H\otimes H^\vee.  If we guess that this product should have a Hodge structure of weight 0, then that tells us that H^\vee should have a Hodge structure of weight -h.  So what happens to (H^{p,q})^\vee? Well, the only natural thing is to call it (H^\vee)^{-p,-q}, and this is exactly the correct answer to make everything work out.

With duals and tensor products under our belts, we can give \hom(H,K) a Hodge structure, of weight k-h, as \hom(H,K)\cong H^\vee\otimes K.  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 \hom(H,K)^{0,0}.  These only give morphisms of Hodge structures of the same weight, though.  The other choice is to take all classes in \hom(H,K)^{r,r} for all r.  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 H be a Hodge structure of weight 2k-1.  For this, we use the filtration.  We have that H_{\mathbb{C}}=F^k\oplus \overline{F^k}.  Quotient by F^k, and then further quotient by the image of the lattice H.  The claim is that this will give us a torus (that there is a complex structure should be clear).  The only question is if H is a lattice of full rank in H_\mathbb{C}/F^k.  The dimensions are rank match up, so we just need to check that this map is injective.  That, in fact, is equivalent to having H\cap F^k=0.  We have this! It’s just because for elements of the lattice H, they are mapped into F^k\oplus \overline{F^k} 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 J_H.  Later, we’ll apply this to the Hodge structure of weight -k on H_k(X,\mathbb{Z}) to construct the intermediate Jacobians of varieties, which carry a lot of (maddeningly mysterious) information about the variety itself.

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