So, today a few lecture series finished up, and I’ve posted notes from some of the older ones in pdfs on my personal website. And here’s the notes, which I’ll admit are getting spottier, as Griffiths is lecturing from his notes on a projector, and Kerr just goes VERY quickly. I’ve stopped taking notes for these two series.
1. Murre 5
There exist varieties over smooth and such that .
In fact, there exists such that in .
The key theorem is suppose that with such that . Suppose there exists a Lefschetz pencil such that . Assume that such that in for general . Then on .
For instance, take a smooth quartic in . Then , and we can take where is a hypersurface of degree . Now, if , then .
Because if , we have that .
On , there are two systems of planes and and . We can check that and so we take , it’s not homologically equivalent to zero.
where and . Then is not homologically trivial on . But is (and I didn’t follow what happened next, looked like a proof that IS homologically trivial on or something)
For sufficiently general quintic hypersurfaces in , .
1.1. Albanese Kernel
Let be a smooth projective irreducible variety over an algebraically closed field . For , we get divisors and is an isomorphism of abelian varieties.
Now, look at . These are zero cycles. We have a surjective map.
What’s the kernel? Let .
Let a surface over with , then .
In fact, it is “infinite dimensional” that is, cannot be parameterized by an algebraic variety.
Equivalently, by such that , then there is no such that is surjective.
1.2. Generalization of Bloch 1979
Let and define is “weakly” representable if there exists such that surjective.
Fix a “good” cohomology theory. If is weakly representable, then .
Recall that , with the image of and the orthogonal complement. Then the theorem implies the theorem of Mumford, because if , then is nonzero, so is nonzero.
Assume there exists such that for all . Then there exists a divisor and two corredpondences in with and such that in for some .
Note: If , and is weakly representable, the assumption if then fulfilled, so this implies Mumford.
If is as before, then .
The operate trivially on for and so must restrict to zero on .
Now, a sketch of the second Bloch theorem.
Take generaic point of . , for . Then (couldn’t read the last bit)
2. Griffiths 2
From his notes.
3. Kerr 2
3.1. Hermitian Symmetric Domains
Let a collection of homomorphisms . a real adjoint algberaic group such that only appear in the representation , Cartan and doesn’t project to the identity in any factor of .
3.2. ID- Hodge Theoretic Interpretation
Let be a vector space.
A hodge structure on is a homomorphism defined over such that is defined over .
Associated to is given by . Then we get a decomposition into such that the eigenvalue of on is .
Now fix a weight . The Hodge numbers and .
Let be the period domain for polarized HS’s of this type,
(I got lost here, and started reading Milne’s notes)
4. Griffiths 3
From his notes.
5. Green 1 – Applications to the Beilinson-Bloch Conjecture
California is like Italy without the art. – Oscar Wilde
Let be a smooth projective variety. There are two ways to look at it. One is to look at it as a compact Kähler manifold with a Hodge metric giving an projective embedding. The other is to look at as already a subset of given by explicit equations and work algebraically.
We use the Hodge metric to get a Hodge structure, which sits inside . A lot of what you get here comes from algebra, but aren’t Hodge structures of varieties, and even occasionally from analysis.
We can take the field of definition of , , to be finitely generated over , by noting that is defined over adjoin the coefficients of the equations. Now, some things we can do with as an abstract field, but others, we can do only when .
Let’s look at . As is transcendental, we can represent elements of this field by where . We can similarly describe . Abstractly, these fields are isomorphic. But they’re different subfields of . We can think of these as both being , which is distinct from .
So look at the elliptic curves and . As abstract curves over , they’re isomorphic. However, over , the Hodge structures are not equivalent.
In mathematics, we’ve got a break between technology, which we need to prove theorems, and intuition, which we need to figure out theorems.
If we write as a field finitely generated over , then where are algebraically independent and are algebraic over it.
Enter geometry. , so for defined over , and if and are birational over .
A point is a geometric point, or a very general point of if does not lie on any proper -subvariety of (that is, the Zariski closure of over is )
For , is very general if and only if is transcendental.
Now, let for transcendental, and look at . For every very general point of , in fact, for , we get a smooth elliptic curve. So we have , and , the discriminant locus, is defined over .
Let be a number field. Then and has minimal polynomial . Now let be the variety defined by , that is, a finite number of points. is defined over , but the points are not, individually, defined over . defined over means that we get complex varieties, one for each embedding into .
Let be defined over and a very general point. Then by then . We can take , so transcendence degree is .
Let be defined over . Take a very general point. Then we get an embedding into , and the field’s transcendence degree is .
Look at hypersurfaces of degree in . If , then if we look at we get a much larger dimensional projective space for , and we hafve the universal family of hypersurfaces.
So, which computations do we actually need the complex embeddings for? Grothendieck learned to compute cohomology groups using just , but for the Hodge structures, really need .
Big Point: Hodge structures require a complex embedding.
Set . If we differentiate, we get , so we have , and this lets us represent the holomorphic 1-form on . Call it .
Now, let be a simple closed homotopically nontrivial curve in . Look at . We can write a bunch of lines, and so we need to integrate along each of these, and , and we need . We can expand as a power series and integrate.
Now, the point is that hte integral lattice is going to depend transcendentally on , on complex embedding .
Now, take a smooth variety defined over . We’ll be wanting to be the Kähler differentials over . That is, the module , for and . We also have where we only have for .
Now, for all actually implies for all . Why? Look at the minimal polynomial of , . Take of this, and the Liebniz rule implies that we have some element of times , and so .
So is a vector space with basis .
These Kähler differentials give us a complex , and need to be careful, this complex doesn’t end at .
6. Brosnan 5
The following are equivalent:
- The Hodge Conjecture holds for all smooth projective varieties
- For all smooth projective varieties , of even dimension and all Hodge classes there exist a and a such that .
There is a similar result in Saito’s “Admissable Normal Functions”
Proof: : If HC holds, then given a Hodge class , there exists another Hodge class which is algebraic such that . “Clearly” for some divisor , therefore .
Suppose that is even dimensional as above and . Then for , we have if and only if , where is the universal hyperplane over .
This is a result of Brosnan, Fong, Nie, and Pearlstein, and of de Cataldo and Migliorini, and uses the decomposition theorem for perverse sheaves.
an admissable normal function implies that for a small contractible ball.
Construction: let be the unit disc, and the punctured disc. Suppose that is a VMHS on . Pick . Then the underlying local system is a -module determined by the action of commuting invertible operators on .
The are quasi-unipotent operators. There exists positive integers such that .
Remark: are quasi-unipotent on implies that they are on , so the monodromy theorem holds for MHS.
If is polarizable variation, then we can find such that the monodromy is unipotent.
6.1. Monodromy Filtration
Let be a vector space and a nilpotent operator, .
There exists a unique increasing filtration on satisfying , for .
Relative Weight Filtration: Let be nilpotent operator on a finite dimensional vector space equipped with an increasing filtration .
there exists at most 1 increasing filtration of satisfying and an isomorphism for all and all .
Deligne noticed in Weil 2 that -adic sheaves coming from geometry always have this property.
Suppose with unipotent monodromy. We say is admissible relative to if:
- is polarizable
- The Hodge filtration extends to holomorphic subbundles of Deligne canonical extension of to a vector bundle on such that the connection has regular singular points.
- If we pick , and use and the monodromy, then exists.
Define the category of variations which are admissible after pullback to make the monodromy unipotent.
Def (Kashiwara “A study of variations of mixed Hodge structure”) suppose that a complex manifold, a closed algebraic set with complement . Then a VMHS on is admissible relative to is whenever we map such that we can complete to , we have admissible relative to .
Then we set the abelian category of admissible variations.
If is quasi-projective, then for any projective completion of .
Define (Saito) for a variation of pure Hodge structure on .
Note: polarizable VHS are always admissible.
Recall the big claim that if is an admissible normal funciton, then is algebraic.
We observe that , so we need an extension where the extension class is to see why we really need admissible.
As an example, take . We define an extension which violates the algebraicity of the zero locus.
Take . Checking carefully, the extension class is , because the Hodge filtration doesn’t extend to a subbundle!
Check that the problem is with extension of , the Hodge filtration.
Let and generated by , and . Then take and , everything and .
Show that relaive weight filtration chosen exists here.
Point: and the problem in this case is the weight filtration.