And these posts are getting shorter, as the talks that I’m even attempting to take notes in are finishing. The other talks are good, they’re just moving very, very fast and given a choice between learning and notetaking, I’m choosing learning. Perhaps once this is done, I’ll make an attempt at posting blogified versions of some of these notes, making them a bit more interconnected, etc. We’ll see, if I do, it won’t be until the Fall, because right now, my big priorities are finishing writing the talk I’m giving on Thursday (my notes will be posted separately from the others) and also working on the lectures for the Math History for Liberal Arts course that I’m teaching (strictly, Ideas in Mathematics, but I’m doing something different with it). Anyway, here are today’s notes:
1. Green 2
Let be a smooth variety defined over where is finitely generated over . Then we defined last time and , and defined the complex
Now, we need to define hypercohomology. Start with a complex , then on .
Now, look at a double complex, and take the total complex. This is a complex if the horizontal and vertical operators anticommute, and then we use , and hypercohomology is defined by .
Note: Hypercohomology is not on the list in “My Favorite Things”
Using Cech cohomology, we define , and add a sign to the Cech differential, then we can take Hypercohomology.
Now, there’s a theorem that tells us that for , .
We can’t read off the integral lattice, though.
is defined to by and that’s .
He compares the derivation of the long exact sequence on cohomology to X-rays…not good for you to see too many times.
Looking at spreads: take , this gives a variation of Hodge structure over , the smooth locus. Looking at the Gauss-Manin connection , we have a complex , and we can get Griffiths Transversality (also called the infinitesimal period relations)
Now, say that , then is generated by .
Recall the example . If we differentiate, but don’t assume that , we get . And now, . And this last term gives us problems lifting. Fortunately, the coboundary map is essentially a measure of how much lifting fails, so the last term can be thought of as an element of , so the whole thing is for some 1-form.
Now we introduct , the codimension cycles defined over and the cycles mod rational equivalence defined over .
So now, we define to be the Quillen th K-group for . Now, in the Zariski topology, we have . Modulo torsion, it’s a theorem of Soulé that we can replace Quillen K-theory with Milnor K-theory.
is generated multiplicatively by elements modulo the Sternberg relations that if for some , then .
(The lecture here became very difficult to typeset notes for, due to a sequence of complicated commutative diagrams. There are printed notes, and I will link to them as soon as they’re online)
The key point is that we have an arithmetic cycle class , and there is a criterion relating Hodge classes, and relating all of this to the Absolute Hodge conjecture.
2. Griffiths 4
No notes taken
3. Schnell 1 – Algebraic de Rham Cohomology and Betti Cohomology
We’re going to be talking about the arithmetic aspects of things. These are the “absolute Hodge classes” and fields of definition.
The basic insight is Grothendieck’s comparison theorem. Let be a smooth quasiprojective variety over , and we have all of the various Kähler differentials.
We define , which is a -vector space.
Note that if , then then we have .
If is defined over then .
Under this isomorphism .
Remark: This is true for any quasiprojective variety (only poles at infinity) and we have two structures on . One is a -structure, the “Betti” structure, and the other is , the deRham structure.
Let be a smooth projective variety over . By Katz-Oda, the Gauss-Manin connection is algebraically defined, and so we have our variations of Hodge structure are algebraic, and we have the SES
the connecting map gives the Gauss-Manin connection.
3.2. Cycle Classes and fields of definition
If is a smooth projective variety over and a subvariety of codimension , then is always a Hodge class.
An important point is that we can also define an algebraic fundamental class .
Let be defined over . Then
In general, use the chen classes of vector bundles with sections vanishing along .
(Detailed definition of chern classes)
Now, we can extend Chern classes to coherent sheaves by resoltuing using locally free sheaves (because is smooth), and for codimension . And so, finally, we set .
4. Kerr 3
No notes taken
5. Charles 1 – Hodge Loci and Absolute Hodge Classes
Let be a compact Kähler manifold. We have Hodge classes .
If is projective, Hodge classes are algebraic.
This is false for Kähler.
Now, let be a field, embeddable in . For smooth and projective over , we have . Now, take an embedding.
We get Betti-deRham isomorphisms which are compatible with the cycle class map.
Let . We say that is a Hodge class relative to is and the image of in lies in the reational subspace.
We say that is absolute Hodge if it is a Hodge class relative to any .
Remarks: First, , we can define what it menas for a class to be an absolute Hodge class. Now, how dependent is this on ? The cohomology classes of algebraic cycles are absolute Hodge.
Proof: If is an algebraic cycle in , then for any , we have algebraic sycle in So this gives a Hodge class relative to , and works for all .
Thus, we have two conjectures:
Hodge classes are absolute.
Absolute Hodge classes are algebraic.
These two conjectures imply the Hodge conjecture.
Let be smooth and projective of dimension . then we have contains a class via the Künneth formula.
It’s a conjecture that the are algebraic.
The are absolute Hodge cycles.
Conjecture: The inverse of the Lefschetz map is algebraic.
This is absolute.
Hodge classes on abelian varieties are absolute.
Let be smooth and projective, an algebraic cycle of codimension in , homologically equivalent to zero. We have , which gives us an extension of mixed Hodge structures .
Now, if and onyl if the sequence splits which is if and only if there exists a Hodge class mapped to 1 in .
This is related to the Bloch-Beilinson filtration on Chow Groups, .
Now, we have out GM connection and a VHS structure.
Remark: If and , and , then . But, this does not for cohomological equivalence!
Indeed, take be an algebraic cycle. Then the cohomology class actually comes from .
For an absolute Hodge cycle, then is defined over .
Proof: We can assume that is finitely generated over , and is the function field of some quasiprojective . We can assume also that extends to , a section of . We want to prove that is constant.
Now, pull everything back to by some . We want to check that is constant.
But, being absolute, for every , is going to lie the rational lattice of , and so will be constant.
a smooth projective morphism with quasi-projective and connected. Let be a global section of , which is Hodge everywhere. Then if is absolute Hodge for some , it is for all .
Just conjugate . for Then .
Now, we need algebraic as a section of . For this, we take the global invariant cycle theorem which says that is the restriction of a Hodge class in where is a smooth compactification.
Application: Let be a HS of weight 2 with . Then there exists a Hodge structure of weight 1, polarized, with
Starting wtih a family of polarized K3’s, get a polarized abelian scheme, then we have for all . connection.