Today the conference ended, and I’ve got an early plane to catch. I’ll be away from the blog for a bit. Here’s the notes from the last day, which, I admit, are a bit sketchier than most of the others, but they’re what I have:
1. Cattani – Mixed Hard Lefschetz and Mixed Hodge-Riemann Bilinear Relations
This is work with Kaplan and Schmid
Started out with quite a bit of review.
Let be Kähler classes, then .
In fact, we can also define primitive cohomology this way, and the Lefschetz decomposition, etc. We get the H-R Bilinear relations this way also, just slightly modified to having the form be .
This is very natural, but it first appears in a 1990 paper by Gromov. He proved it for (1,1). It also appeared in 2006 by Dinh-Nguyen.
In between, there was work on cohomology algebras for polytopes, by Timorin.
Now, , and now, being purely primitive doesn’t guarantee the sign. Look at , has signs .
We care about things like this to prove things about intersection cohomology.
Now, look at some integral polytopes. They give and we set to be the common zero locus as a subset of
Then BRK proved that is the number of points in .
A polarized Hodge-Lefschetz module of weight is the following data:
- a -graded vector space over ,
- a nondegenerate bilinear form of parity
- and an abelian subspace
- There exists a bigrading with and
- for all
- For , is an isomorphism
- For , the induced HS on is polarized by
Let be the polarizing cone .
Let for a compact Kähler manifold, and .
Let , and set and , , and let the Lie algebras act on this.
This data is a PHLM of weight and bigrading .
This is a PHL module, and every PHLM arises in this way.
2. Cautis – sl(2) actions and Hodge modules (on Grassmannians)
2.1. Basic sl(2) review
Let be a rep (finte dimensional). It breaks up into eigenspaces (the weight spaces) , , through . with and the raising and lowering elements. If we restrict to it is just for some .
Remark: if , it gives an isomorphism .
2.2. Categorical sl(2) actions
The data consists of additive categories with functors such that such that on , we have , and .
Take as varies. That is, teh category of mixed Hodge modules on grassmannians, and set . Then given the collection of flags , we have forgetful maps to and , and we get and by pulling back and pushing forward along these maps.
We claim that this is a categorical sl(2) action.
For , we compute (by convolving) that is given by taking the collection of flags with the in common, and forgetting it, getting a subset of .
Then, is given by the intersection cohomology, and this is an simple application of the decomposition theorem.
Given a “strong” categorical -action, one can form natural transofmrations, and induces an equivalence of homotopy categories.
There is an iso
In this case, this equivalence is given by where is the open subset where we have .
In fact, gives the functor , and the inverse is !
2.4. Associated Graded
Look at the filtered module of differential operators on . MHM’s are filtered modules in a way compatible with this. Now, .
Now, if , then is a -module, which is also an object of .
The idea is to use the fucntor to get the action on .
(I stopped understanding it around here)
3. Migliorini – Topology of Hitchin systems and Hodge theory of character varieties (joint work with M. de Cataldo and T. Hausel)
Let be a nonsingular projective curve .
The Narasimhan-Seshadri correspondence (special case, rank 2 and degree 1) is and be the moduli space of stable rank 2 degree 1 holomorphic vector bundles on . There exists a natural real analytic diffeomorphism .
Hitchin complexified this. Let , and the moduli space of stable rank 2, degree 1 Higgs bundles on . There exists a natural real analytic diffeomorphism .
Now, is affine, but is covered by mid-dim compact algebraic subvarieties.
Hitchin’s map takes to , where is the sum of the spaces of abelian and quadratic differentials.
Also, has sympletic holomorphic, and a action .
The map is proper (in fact, projective!), is -equivariant, and is a Lagrangian fibration. To find the fibers of , take , and then look at the spectral curve over given by .
If is a rank 1 torsion free sheaf on , then is a rank 2 locally free sheaf on , so
, the compactified Jacobian.
Back to , which is a smooth affine variety. MHS on studied by Hausel and Rodriguez-Villegas, has the property that it is split over and of Hodge-Tate type. They also discovered “very strange symmetry” which is a “Curious Hard Lefschetz”…that is, there exists a class of type , such that is an isomorphism.
Then, there’s the nonabelian Hodge theorem, which says that and this gives a pure HS on . But also, defiens several filtartions on (Leray filtration, and modificulations ) and it was introducted in BBD by setting .
Now, associated with a spectral sequence degenerating in the page, and if the map is smooth, it is the leray filtration. Also, if is projective and relatively ample, then .
Cohomological characterization of : If projective, and assume is affine, then we have the generic dimensional linear subvar of .
Under the NAHT, the weight filtration on is sent to on . The plitting of corresponds to Deligne’s splitting of .
The same result holds for and , but much harder
What about rank 2 and degree 0? The guess is the that same holds for intersection cohomology. What about other groups? ? How good is this?