Here’s the second set of notes, the beginning of the crash course in Hodge theory before we get to advanced topics next week. My notes from the first day and a half are available here, please let me know if there are any errors or any bad code that should be fixed. Without further ado, Hodge Theory, day 0.5:
1. ElZein 1 – Hodge Structures and Mixed Hodge Structures
Work of Deligne, then Griffiths
Hodge decomposition is a geometric invariant: this means that if is a morphism of compact Kähler manifolds, then we have a map , we have and . If is analytic, then we have .
The Hodge decomposition is a linear structure on the cohomology.
The cohomology groups of algebraic varieties carry mixed Hodge structures.
A hodge structure of weight is defined by a finitely generated group , a decomposition of into a direct sum where is a complex subspace such that .
Let and . We can make a Hodge structure by and , but not by trying to make .
1.1. Hodge Structures of Weight 1
, then . We have gives an isomorphism , and the image gies a lattice, so is a torus.
When is Kähler, , we have an exact sequence of sheaves using the exponential map which gives a long exact sequence including where the last map takes line bundles to the first Chern class. So then is the kernel of Chern map, and this is a complex torus called the Picard torus.
1.2. Algebraic Operations on Hodge Structures
If and are Hodge structures of the same weight then so is .
with is a Hodge structure of weight .
If and are Hodge structures of weight and , then is a Hodge structure of weight
is a Hodge structure of weight
Take and of weight .
We define , and this gives a Hodge structure of wieght with .
A Hodge structure of weight is defined by and a finite, decreasing filtration by subspaces such that for all , .
The two definitions of Hodge structure are equivalent.
Proof: Start with a decomposition. Define . Then , so the property of filtrations follows. For the other direction, define .
A morphism of Hodge structures is a map defined on the abelian groups such that, after complexification, satisfies .
A polarization of a Hodge structure of weight is a bilinear form which is symmetric for even and skew-symmetric for odd such that the complex extension satisfies unless and , and that for .
This gives a positive definite Hermitian form.
A mixed Hodge structure is defined to be a finitely generated group , an increasing filtration on and a decreasing filtration such that ‘s complexification has a Hodge structure of weight induced from .
2. Cattani 1 – Odds and ends
2.1. Riemann-Hodge Relations
Let . We have , and we define a number using , then the Riemann-Hodge relations say that this gives a polarization on . We’ll denote this by .
In dimension one, we have , and then , and if .
Subset: , example 1.16.
Look at the sequence , it gives the long exact sequence with the maps , with the last map taking to .
We have and , and with the map taking to .
Question: Can we find, in some natural way, 1-forms such that ?
Let be a line bundle, such that and .
Taking , we get , and to get we just exchange and .
Suppose you have a coframe, that is, a basis for sections of , . We want a derivative map . We need , then we’ll have where .
These will need to satisfy some condition under change of coordinates in order to make sense. If is the matrix of transition functions, then if . For any individual we have .
Finding such that is the same as having a connection.
3. Migliorini 1 – Hodge Theory of Maps
The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar constraints on algebraic maps?
Let a proper morphism. Our goal is going to be to linearize the problem.
For this lecture, we will assume is projective and smooth, to simplify the problem. In fact, we will assume and are nonsingular.
So our map factors and is surjective. Equivalently, there exists a line bundle on which restricts on every fiber to an ample line bundle.
If is a projective, nonsingular variety, tehn there’s a universal hyperplane section and call it , we have a map . This is not smooth, and we call this map . Define to be the set of hyperplanes tangent to . Then , we have a smooth projective morphism.
Lemma 1 (Ehresmann Lemma) Let be manifolds, connected, a proper submersion. Then all of the fibers of are diffeomorphisc and is a locally trivial fibration.
In fact, if is contractible, then is just .
If is a look in starting at , then you get in general a diffeomorphism of to itself that is not isotopic to the identity.
And example of this is to take a family of elliptic curves over , then a nontrivial loop in gives a nontrivial (nonisotopically trivial, even) diffeomorphism .
As the diffeomorphisms depend only on the homotopy classes of looks, we get a natural representation .
Let . We define to be the sheaf on associated to the presheaf which sends to . These are locally constant sheaves, but not necessarily constant, due to monodromy, that is, the representation of mentioned above.
So what does all this have to do with Hodge theory?
Consider , the Hopf fibration. Here .
Also look at . Then . The Künneth formula tells us that . But for the Hopf fibration, this fails: is different.
In general, for a fibration, knowledge the the does not suffice to compute the cohomology of the total space.
Now, if is a fibration, there is a Leray Spectral Sequence with .
If is projective and smooth with connected, then the Leray spectral sequence degenerates at .
If is simply connected, then , as groups.