The summer school is over. Tomorrow begins the conference, including talks from such luminaries as Arapura, Illusie, Griffiths…and that’s just the first three talks tomorrow. I’ll be talking on Thursday at 17:05, and until then, it’s polishing time for my talk, and working on a new lead on the research it’s expositing. Meanwhile, today’s notes are somewhat sparse. I’d lost track of the Shimura Varieties thread days ago, and today I lost the Beilinson-Bloch thread, sadly. Hopefully I’ll work it out later, but my lack of arithmetic background has caught up with me, I couldn’t keep everything straight…well. Here’s the notes from today, tomorrow’s should be more modular:
1. Kerr – Deligne’s Theorem on Abelian Varieties, Part II
Let is a CM abelian variety, that is, an abelian variety such that is abelian.
Now, if and then we want to show that lives in .
Let be a CM field of degree such that is totally imaginary and there exists with , for all .
Now, take to be the totally real fixed field, and such that , and and for with generated by . We call the CM type of .
Now, consider an abelian variety with . Then is an -vector space of even dimension and .
Now, is self-dual, and so acts on and we have natrual quotient map , and the dual is an inclusion defined over .
is a -vector space of dimension and it acts on and similarly for , and we have
The HS on may be viewed as taking to the -linear endomorphism of multiplciation by on and on and this must commute with .
Therefore, , and are of dimension and with .
So the Hodge type of is .
Conclusion: If for each , then consists of Hodge classes (the Weil classes).
If is an abelian variety of dimension and times, this is then . Let , this is just , and so taking to act on the factor of , we get .
This gives us that .
Moreover, changes neither the product structure on , the endomorphisms (which are defined by cycles in ) nor the class of on . Thus, in this cases consists of absolute Hodge clases.
Now, think of as a fixed -vector space of dimension with nondegenerate alternating form .
Let be any weight 1 Hodge structure on polarized by and an isomorphism (in such a way that gives and . We impose the condition that for all .
Then there exists a unique -Hermitian form with and stabilizes and commutes with . Hence, and , is a MT domain which precisely classifies the abelian varieties (or HS’s) satisfying the above conditions which are precisely that the HS for which consists of Hodge classes.
Now, a torsion free congruence subgroup is by the Baily-Borel theorem a quasi-projective algebraic variety parameterizing such .
Applying Principle B again leads to
Weil classes on “Veil algebraic varieties” are absolute Hodge
The rest of Deligne’s proof: Let be cut out by and be cut out by (the Hodge and absolute Hodge tensors) then
If a tensor is fixed by , then it is absolute Hodge.
For CM abelian varieties, Deligne shows that is an equality by producing enough absolute Hodge classes to push inside . He does this by looking at endomorphisms of the CM field, and Weil Hodge classes.
This is dense on .
2. Green 4
Today we’re going to look at cycles over on a variety defined over .
Let be defined over . Then is captured by cycles classes and .
That is, .
The Conjecture in fact says that for the transcendence degree of , plus two, for defind over .
Now, if is defined over and a finitely generated extension of , we can find with , and set , and for any cycle we can spread it to , and there will exist a a proper subvariety, defined over of lower dimension with such that .
If the conjecture on varieties over is ok, then in rational equivalence over for some , so is torsion, and so its Abel-Jacobi image is also zero after tensoring with .
(I lost track of the lecture here)
3. Kerr 5
No notes taken
4. Green 5
No notes taken