Today was the penultimate day of the summer school. Once it finishes, there will be a conference, and things go through Friday, so all-in-all, there will be 15 days in this series. And here’s today’s notes:
1. Griffiths 5
No notes taken
2. Charles 2
Let be a smooth projective (maybe quasi-projective) variety and of characteristic 0 and embeddable into .
Fix and , say that is Hodge with respect to if with the isomorphism then lies in the rational subspace and .
We say that is absolute Hodge if is Hodge with respect to every .
Now, if and is absolute Hodge if is is a hodge class and its image in is absolutely Hodge, so Hodge with respect to all .
Let be smooth, projective over with quasiprojective, smooth and connected. Let be a global section of such that is of type at any point of and for some , is absolutely Hodge. Then , we have that is absolutely Hodge.
Remark: If we know is algebraic, then we can prove is.
We prove this before.
Let as above, the monodromy representation. Then the fixed points of are those in the image of where is a smooth compactification.
In our proof, is a global section of , and so it is -invariant and thus comes from , so is algebraic. Then , we have that is absolute Hodge.
Remark: if is defined over , then Hodge classes on are always absolute Hodge.
Question: How can we reduce questions in Hodge theory (Hodge cycles are absolute, Hodge conjecture) to questions over and number fields?
Let be smooth projective. We can always find with smooth and projective, smooth and quasiprojective and such that ( corresponds to a point of some Hilbert-scheme of subschemes of )
What does it mean for Hodge classes fibers to be absolute?
Remark: , so the Hodge bundle are defined over , and so is .
If , and permutes the complex points of the Hodge bundle, then
is an absolute Hodge class if and only if is rational and is an absolute Hodge class for all .
The locus of Hodge classes for is the set of such that is a hodge class.
Hodge classes for the fibers of are absolute iff the locus of Hodge classes is invariant under .
The locus of Hodge clases is actually a countable union of analytic subvarieties of .
Start with . We get a component of the locus of Hodge classes by taking parallel transport of over some small open subset and looking at the vanishing on .
Lemma 1 Let be defined over and be a countable union of analytic subvarieties such that is stable under the action of . Then is a countable union of algebraic varieties over .
The idea is to take a very general point of and look at the orbit of it under .
We know the geometric part of the conclusion:
Let as before. Then the locus of Hodge classes in is a countable union of algebraic subvarieties.
We don’t get information on the field of definition.
Let as before. The Hodge locus of is the image in of the component of the locus of Hodge classes passing through . By DCK, this locus is algebraic. Let be in this locus. Assume it is defined over . Then the Hodge conjecture for can be reduced to the HC for some .
Proof follows from DCK and the global invariant cycle theorem.
3. Green 3
Remember, we have finitely generated, and this gives us with . We also start with defined over and get .
We set the cycles on defined over , take it mod rational equivalence on defined over , and we get .
There is a conjectural filtration with the homologically (but not rationally) trivial cycles.
For any we get a map which should take to .
. Let . Then , and then if and only if .
Now look at . We expect that is the kernel of , the Abel-Jacobi map for . So we need that this kernel doesn’t depend on which is chosen for very general .
Beilinson’s Conjectural Formula: where is the category of mixed motives over . So, in particular, we have no idea how to compute this group in general.
Now, we’re going to talk about and the Abel-Jacobi map.
Let be a smooth projective variety defined over . Take smooth cycles defined over , the inclusion, defined over . Set . Then we have and we have differential .
Then , so we have a complex.
We can get an exact sequence , and we can construct splittings that respect the Hodge filtration, and splittings that respect the integral lattice, but not one that does both! So, working out the set of extension classes, we get , so it is .
4. Kerr 4
No notes taken
5. Schnell – Deligne’s Theorem on Abelian Varieties, Part I
On an abelian variety, all Hodge classes are absolutely Hodge.
The proof breaks up into two parts:
- reduce to the case of CM abelian varieties
- Deal with CM case.
We’ll deal with step 1 today.
Recall, in the case of weight 1:
A CM field is a number field of the form where and is totally real and under all embeddings , is negative.
An abelian variety is CM if there exists a CM field such that
This implies that .
There is a nice criterion (MT means Mumford-Tate group)
If is simple, then is CM if and only if is abelian.
Given any abelian variety and a Hodge class on , there exists a family of abelian varieties with irreducible and quasi-projective such that there exists with and the Hodge locus of is , and there is where is CM.
Proof: Choose a polarization and let and the smallest -subgroup whose -points contain the image of .
Abelian varieties of the same kind, along with a choice of basis for , are parameterized by the period domain .
Note: points of are classes of in terms of . .
Main idea: family comes from the Mumford-Tate domain: .
This should have the properties that for all Hodge structures ,
- any Hodge tensor for is a Hodge tensor for
- for .
Finding CM points corresponds to finding points with abelian MT. contained in some maximal -toruc , and we can show that for generic, is the stabilizer of .
Nearby, there exists close to , then if is the stabilizedr of , it is a -torus. There exists such that , and has image in . Then is abelian.
Problem: family over quasi-proj base, not . Solution: Fix an and use a level structure.
Define to be the moduli space of abelian varieties of dimension with polarization and level structure (a basis of the -torsion points) and let the universal family. OUr replacement for is to let be the Hodge locus of the Hodge tensors for defining .
is algebraic by CDK, and finite etale over . In this case, things are ok.
Proof that (for simple), abelian implies is CM.
We start with the fac tthat is a division algebra, since is simple. It is also the sety of -endomorphisms that commute with . So we know that is abelian, and thus it acts on , and we can write for characters, and thus .
And so is bounded above by . So and thus is a commutative field, so .
Now, use teh Rosati involution on , and , and the fixed field. We claim that and is totally real.
We have that , with and take the minimal polynomial. Then , the roots, are the eigenvalues of the action of on , and if we set , and acts on preserving , there exists with , . Look at , this is and so , so .
6. Griffiths – Colloquium – Hodge Theory and Representation Theory
Joint work with Mark Green and Matt Kerr
Lecture went faster than I could type. No notes.