Sorry these are going up a bit later than usual, had some difficulties last night, and was working on my talk. I’m just not taking notes whenever there are slides or lecture notes on the overhead, because it’s impossible for me to keep up while taking notes on those talks, so that will limit these a bit, but here are the other talks:
1. Arapura – Beilinson-Hodge cycles on semiabelian varieties
Joint work with Manish Kumar.
Reminder: If is a smooth projective variety over , then there’s a cycle map .
This map is surjective.
We’ll abbeviate the Hodge Conjecture as HC.
Now, assume smooth but not necessarily projective, so open (quasiprojective). Then the cohomology carries a MHS.
We define the Beilinson-Hodge cycles, and then the Hodge cycles correspond, in the projective case, to .
Do these cycles have a geometric origin?
We should make this rigorous, so we look at Bloch’s higher Chow groups. Try the singular homology.
Look at , and we can look at as embedded to contain the hyperplane .
So we replace with cycles on satisfying that the ?? is in the expected dimension.
So we get a complex and we define to be the th homology of this complex.
The usual Chow groups .
, the group of units in the ring of regular functions on , because .
is the motivic cohomology, though we’re going to focus on the concrete cycles.
Given a cycle , the fundamental class of gives a class , and if we pass to topology, we can explain this as and then we integrate out the sphere componenet to get . And it turns out, as MHS, we have .
So we have a cycle map.
This map is surjective for all .
We’ll call this .
If is smooth and projective and , then this is just HC.
BHC is false.
This can be seen in the case and , and is related to Mumford’s example of a surface where is different from .
However, the conjecture if believed to be ok if is defined over , and the work of Asakure-S. Saito suggests that it seems ok so far for .
Suppose that is a product of smooth affine curves or semiabelian over . Then holds for unconditionally, and if HC holds for a smooth compactification , then is true for all .
Suppose that is as before, but defined on a finitely generated field . The cycle map is surjective where and if the Tate conjecture holds, then a similar statement holds for all .
The above can be considered as a Beilinson-Tate conjecture.
Idea of the Proof of Theorem 1
Just as the HC has the easy case of the (1,1)-theorem, there’s an easy case of the BHC.
Recall that , then the cycle map is given via the exponential map.
Lemma 1 (Abel) The map is surjective onto the Hodge cycles in . That is, is true
In fact, is a ring, and likewise, is. The Mumford-Tate group is isomorphic to the Galois group.
Now, given a -MHS , then there exists a -algebraic group such that is equivalent to the tensor category generated by .
We’ll take a subgroup the Special Mumford-Tate group .
leaved Beilinson-Hodge cycles on tensor powers of invariant, and conversely.
Note that is generally not a reductive group, but we need to do invariant theory for it, and this reduces the theorem to .
2. Illusie – Semistable reduction and vanishing theorems revisited, after J. Suh
This talk is dedicated to the memory of Eckart Viewhwig
This talk was on slides, and I couldn’t keep up.
3. Griffiths – Hodge domains and automorphic cohomology
Talk based in part on joint work with Mark Green and Matt Kerr
From the early days, there was a mystery about the relation, if any, between automorphic cohomology and variations of Hodge structure . Recent work of EGW and C and on Mumford-Tate groups and domains have shed some light on this issue and we will discuss this and related manners.
Here, is a Hodge domain (period domains are special cases), is a weight in it with Harish-Chandra character for the corresponding discrete series representation in . a line bundle on in Schmid’s construction of the automorphic cohomology ralizing the discrete series representation with character where , and is a smooth quasiprojective variety where is a NCD with unipotent monodromies and is a locally liftable holomorphic mapping with where si is ???
(here, the slide changed faster than I could type)
Classicaly, is a hermititan symmetry domain and automorphic forms give a map between representation and algebraic geometry.
The question is, what is the relation between these in the nonclassical case?
In the very classical case, that is, the weight , and parameterize the complex points of a component of a Shimura variety, there is an additional rich an extensive arithmetic aspect and one has relateions between algebraic geometry, arithmetic and representation theory via variations of Hodge structure, discrete series and cuspidal automorphic representations, and L-functions and Galois representation.
Recently, it has been found that the -algebraic semisimple whose have discrete series are exactly the Mumford-Tate groups that arise in Hodge theroy. And these are the that are hoped to have cuspidal automorphic representations in .
The use of cycle spaces and their enlargements has given a way to map to the (slide changed too fast)
3.2. Hodge Groups/Domains
Let be a -vector space, a polarization and and the maximal compact subgroup of .
A Hodge structure is and a polarized Hodge structure is a map (slide change)
(The slides changed too quickly for me to keep up after this)
3.3. Cycle Spaces and their enlargements – Examples
3.4. Enrose-Radon transforms
4. Green – Vanishing of Chern Polynomials for Hodge Domains
This is joint work with Carlson and Griffiths. It has appeared in the online journal SIGMA.
Let be a period domain, we have a period map. Griffiths discovered the infinitesimal period relations, which to a differential geometer is a colletion of 1-forms which is not an integrable exerior differential sysem.
But, for any VHS, we have the integrability conditions.
On the period domain, we have the bundles from the Hodge filtration , and the quotient is , as we’re only looking at polarized Hodge structures of weight , with given Hodge numbers.
So we have where is a flat bundle. Now, because we have , naturally. This, of course, gives relations on the Chern classes, because , but this isn’t what we’ll be talking about.
We also have that is conjugate to .
Now, let the natural map, then the IPR is that and .
So we get , but this is just .
So we look at and dualize, and we het , and this is just , and the IPR says that for all , .
For the integrability conditions, we map , and the integrability condition is that . (Here we mean that we compose and wedge)
This always holds for VHS.
The integrable tangent space is then .
We have a map and the curvature of is with columns and and similar rows, and the block is an matrix of -forms.
To compute the Chern forms, say, by Chern-Weil, we look at , with by convention.
Restricted to any integrable , if .
So we have (well, in the nonzero block), then .
Now, is in the nonzero block , and in the nonzero block.
Now, I want to multiply these together: , we get (in the nonzero block, of course) , but , and so the product is zero when restricted to !
So how can we make use of this?
Notation: If is a matrix, set .
If are matrices such that , then for .
Note: We’re not working over a field, if we were, then this would all be obvious.
Over a field, implies , and if .
Proof: We want ot look at . This is also , and as , every term has degree , because everything has a or . Thus, if of degree less than , which is the same as .
5. Schnell – Neron models and Poincare bundles
Let be a polarized HS over of weight and which the polarization makes isomorphic to .
The points of correspond to extensions of MHS, . this gives with a splitting, which we call , and by dualizing we get which gives us that , and classifies the extension.
Problem: Given a family of intermediate Jacobians that degenerates, extend it in a good way (Green-Griffiths).
Let open and a -VHS, polarized of weight , then we have .
Why do we want to do this? Because we want to study the behavior of normal functions on .
Let be smooth and projective, . Then we have a family open in which is a VHS, and the variable part, and we have .
Green-Griffiths-Kerr deals with the case where . This was generalized by Brosnan-Pearlstein-Saito who gave a more topological construction, and there’s also wrok by Katz-Nakayama-Usui.
So now, and it is natural to extend and then take the quotient by .
and , then a VHS is the same as a nilpotent orbit.
We can -split a polarized MHS, and use the Deligne decomposition with conjugate to and and .
This allows us to construct a VHS on with monodromy and the period map is on , it will take to .
One way: has a canonical extension with , and the sections of come from for , and is generated by for .
So what happens to the integral classes?
We take a sequence over points which go to such that converges for all . Now, we can rewrite this as , and is just .
This imples that converges and that converges.
This is not enough to control individually, though.
Fix: Suppose we also allow derivatives . This is just , and in fact we get for . Test on this, and converges. Call our three sequence .
Solve: , and this is just plus conjugates, and .
Let the inclusion, then we have that along with is a D-module, and we can take a sub-D-module generated by , and call it .
We set , then , etc, and this leads to the “minimal extension”
General: consistent use of Saito’s theorem.
Let be the category fo mixed Hodge modules on . An object here is a perverse sheaf and a filtered -module .
An example is a VMHS, with and .
Construction: a polarized -VHS on of weight -1, , then take an integral extension , which gives with a coherent sheaf on .
and has section . Then is the etale space of , and finally, there exists taking to .
is closed analytic embedding, then is a Hausdorff-analytic space, the “Neron model”
- Any admissible normal function without singularities extends to a holomorphic section of which is horizontal.
- For all ANF, has analytic closure on .
is algebraic if is algebraic.
Interpretation for singularities: ANF has a singularity at if and only if it does not extend to a section.
Now, we look at Poincaré bundles.
Green-Griffiths: try to construct a Poincaré bundle on and use it to get invariants of ANF.
What can we do? Start from an ANF for without singularities, then construct a line bundle , and given , we construct on .
6. Pearlstein – The locus of the Hodge classes in admissible variations of mixed Hodge structure
This is joint work with Brosnan and Schnell
Let be a MHS with lattice , then a Hodge class of is an element of .
Let be a VMHS with lattice , let the etale space of .
. A choice of graded polarization , where is the locus of elements with .
Let be a Zariski open subset of a complex manifold and be an admissible VMHS wrt . Then extends to an analytic space, finite and proper over .
If pure and weight 0, this is due to CDK.
If is a quasi-projective variety, then is also quasiprojective.
Let be an admissible higher normal function on , ie, comes from where is pure of weight . Then the closure of the zero locus of in is complex analytic.
If is quasi-projective then is algebraic.
Let be an admissible VMHS on a quasiprojective variety , then the locus of points where is split over is algebraic.
WLOG, assume that each is defined over , and is torsion free, connected. Replace by .
A generalized normal function on is an admissible extension where is a -VMHS with weight .
Let be an admissible generalized normal function on then the closue of in is complex analytic.
Proof: Let be the corresponding VMHS. If is pure, we’re done by theorem 3. Otherwise, let be the smallest integer such that .
We define and the associated generalized normal function. , and by induction, we can assume is complex analytic.
Let denote the regular locus of an irreducible component of , then which is an isomorphism over . is proper implies that we can replace by and by the pullback to , so we can assume .
, get and pure.
Theorem 3 implies that has analytic closure in , so .
Now, we prove Theorem 1:
Let be an admissible VMHS, . For any point , we have implies that extends into . Let an irreduciblecomponent and irreducible and containing .
extends to an analytic space which is finite and proper over . We have with . Pull back to , and call it .
By construction, we have a section , and so we get a generalized normal function where .
So and then by the proposition, we have which is analytic.
7. Movasati – Automorphic functions for moduli of polarized Hodge structures
No notes, due to fast moving slides