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.

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About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
This entry was posted in AG From the Beginning, Algebraic Geometry, Cohomology, Hodge Theory. Bookmark the permalink.

2 Responses to Hodge Structures

  1. Steven Sam says:

    For the dual, isn’t it that hom(H,K) is isomorphic to H^* \otimes K and hence has weight k-h?

  2. Ack! You’re right. Write everything backwards there. Corrected.

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