## Hodge Structures

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.