Today we kept going with the advanced topics, and started Shimura varieties, though that one went VERY fast. As always, comments and corrections are welcome!
1. Carlson 3
Let , the discriminant and the fiber over where is nonsingular.
Let’s look at the example then . Set to be the flow where is a diffeomorphism. Then set , and so we get the monodromy representation, and .
For odd it preserves a skew-symmetric form, for even a symmetric form, so we have or .
And this gives , and we can descend this to the quotient where is at least as big as the image of the monodromy, at at most .
To get a specific example we can compute, let’s look at the family . Let . Following around this loop, we see that its image is , and it’s a Dehn twist.
For the family of cubics, we have a map , in general, the image of is large.
Beauville showed that for hypersurfaces, is surjective or has finite index.
1.2. Asymptotics of the Period Map
Let . Recall that , and and as yesterday. We want to do the asymptotics of as , we get , and for we take and get , and this simplifies to by using trig identities.
So the -period is and the -period is , and the period map is then .
. For algebraic surfaces, . In general, weight of the Hodge structure.
1.3. Period Domains and Hodge structures of weight 1
First, note that the number of moduli is unless , where it is 1.
An abelian variety is a complex torus thatis a projective variety.
The moduli of abelian varieties of dimension is .
Now, given a weight 1 Hodge structure, is a projective variety, by the Riemann Bilinear Relations and the Kodaira Embedding Theorem.
The Torelli Theorem says that the map is injective, so we get a map , and finding equation for this image is called the Schottky problem.
What is ? In the general case of weight 1. We write down a matrix of -periods and -periods, with the basis a basis for and a symplectic basis for .
We define and . The Riemann Bilinear Relations tell us that for a nonzero abelian differential, and .
Putting this all together, we get that we get that , which is just , and nonsingular matrices. So we can find a basis such that the period matrix is . The first RB relation gives that .
So, for weight 1, .
2. Murre 3
Let be a smooth projective irreducible variety over , there are several topologies, in particular the Zariski topology and the étale topology, as well as teh underlying analytic space which is a complex analytic manifold which is compact and connected in the classical topology.
We have a comparison theorem saying that for the characteristic.
Similarly, we have a theorem of Serre in 1956, GAGA, that is an equivalence of categories between algebraic and analytic coherent sheaves, and the cohomologies are the same.
In particular, we find that the Picard groups are isomorphic, and so doesn’t depend on whether we look analytically or algebraically.
2.2. Cycle Map
Let with .
There exists a homomorphism given by taking the inclusion and using the exact sequence of a pair for the analytic spaces.
In particular, we have maps and , and . Taking to an element of , we define .
What is the position of ?
Lemma 1 Let then , and hence .
Proof: We have a nondegenerate pairing between and . Set , and take a . Look at , and this is zeor unless .
Reference for details: Griffiths and Harris, or Voisin.
2.3. Hodge Classes, Cycles and Conjecture
We have the theorem that the image of is contained in .
This is true for , by the Lefschetz (1,1) Theorem. But for , this is no longer true! See Hirzebruch-Atiyah in 1962, Kollar in 1992 and Totaro 1998.
However, it is still wide open if it is true after tensoring with .
But we do have:
Sketch of proof, following Kodaira and Spencer:
Here we have that , and we have the exponential sequence exact, so we get an exact sequence on cohomology. This gives us an exact sequence and so we get that , and every (1,1) class has image zero in , so the map is surjective onto them.
2.4. Intermediate Jacobians
Let be the th intermediate Jacobian of over , then . This is the Griffiths intermediate Jacobian.
Lemma 2 is a complex torus of dimension .
Proof: To see if is a basis for , we need . We have that the , so , so , and .
This is in general not an abelian variety. However, if , then it is.
This will always happen for and where , and these are the Picard variety of and the Albanese variety of , and, in the case of curves, these are the same and are the Jacobian of of .
3. Brosnan 3
Last time, we looked at is the Griffiths intermediate Jacobian for pure of negative weight, by Carlson’s formula.
For a smooth projective variety, which has weight -1.
Recall from Cattani that a variation of Hodge structure of weight on a complex manifold consists of a pair where is a local system on , is a decreasing filtration on and at every point , we have defines a Hodge structure, along with Griffiths Transversality, that .
Now, Griffiths notices that if is a smooth projective morphism, then if you set so that , then we have a variation of Hodge structures.
A variation of MHS is a triple , with a local system, and appropriate filtrations, see Cattani.
And, just to be clear:
A local system of modules on for a ring is a locally constant sheaf of -modules. If is connected, then local systems of modules are in bijection with modules.
Let be a complex manifold, a variation of pure Hodge structures on . The group of normal functions on is the group .
In this, denotes teh abelian category of variations of MHS on and denotes the constant pure Hodge structure on which is at every point, and the weight of is negative.
Given , we have a family of intermediate Jacobians and the fiber at is , so by restriction, an element determines a section of .
Fact: determines . In other words, there is an injection to the group of section of a family .
The idea of hte proof is to try to use the section to construct an extension of mixed Hodge structures. What would be nice would be to have a tautological extension of by sitting over . Then you can pull it back by and can ignore Griffiths transversality and find such a tautological extension to get injectivity.
Look at and , then , and we recall from last time that we constructed an element corresponding to , and we gave it as an extension , and this is in , which is isomorphic to .
Suppose that open immersion of complex manifolds, and a VHS on . Typical situation: with smooth projective varieties, and by generic smoothness, there is a dense oepn where is smooth. Therefore, if we set , we get that is a variation of Hodge structure on , but won’t in general extend to .
Take then is a topological invariation of , the singularity of at a point . We define it to be which captures the topology of near as follows: take a small ball around in . Defien to be the image of under the map . As long as is small enough, this is independent of the ball.
4. Carlson 4
Just a few more things about weight 1 before going on to weight 2.
. Another way to view it is that . We can see this by noting that acts transitively on , and the subgroup fixing is .
More generally, has a transitive action by . Let be the isotropy group for , then , and this is an example of a hermitian symmetric space, though higher weight period domains generally aren’t.
4.1. Higher weight
Let be a Hodge structure of weight , the period domain, then is or , and acts transitively on , and let be the isotropy group of
In the weight 2 case, if and , then and with , the maximal compact, .
is not in general a Hermitian symmetric space, but it is when . For instance, if .
For instance, K3 surfaces.
Fact: is compact complex subvariety of .
Period map for hypersurfaces has a holomorphic part and a horizontal part, and .
Poincaré Residue: Look at the cohomology of . Grothendieck looked at this cohomology and its meromorphic forms with pole along .
Let be homogeneous coordinates on , let be , then we have affine coordinates by dividing by , and so . Set the numerator as .
Define where is the equation of with , so then is acohomology class in .
We have a sequence so we define the local residue map in the usual way.
Claim: The period map is both holomorphic and horizontal.
To see that it’s holomorphic, let be given by . Then , so and by the residue formula, the map is holomorphic. Horizontality then follows from Griffiths Transversality.
5. Kerr 1 – Shimura Varieties
The plan of the course is:
- Hermitian Symmetric Domains,
- Locally Symmetric Varieties,
- Theory of CM
- Shimura Varieties
- Fields of Definition
5.1. Algebraic Groups
An algebraic group defined over is an algebraic variety with morphisms and along with subject to the rules making a group for all .
is connected iff is irreducible and is simple iff nonabelian with no normal connected subgroups other than the trivial ones.
If , then examples are the usual algebraic groups, , over , we have the real forms, and over “all hell breaks loose”
We call a torus if is a product of ‘s.
Inside , there is , being the matrices with , the next being and the last being .
Let be Weil restriction. Then , and .
We call semisimple iff it is (almost) the direct product of simple groups, and reductive if linear reps are completely reducible.
One representation is where is the Lie algebra, by
For semisimple , we say that is adjoint if is injective, and simply connected if any isogeny with connected is an isomorphism.
Let be a reductive real algebraic group and an involution.
Cartan if adn only if
This holds iff for some with , such that on .\
if adn only if compact.
5.2. Three Characterizations of Hermitian Symmetric Domains
Let be a connected open subset of with compact closure, such that acts transitively and contains the symmetries . We call it a bounded symmetric domain, and these are analytic and extrinsic objects.
Hermitian symmetric spaces of noncompact type (analytic, intrinsic) are a connected complex manifold with Hermitian metric such that acts transitively, contains symmetries for all , and is semisimple adjoint and noncompact.
The third type, called circle conjugacy class, is a conjugacy class of a homomorphism in an algebraic group over where is a real adjoin algebraic group and only appear as eigenvalues in the red og on , is is Cartan, and doesn’t project to 1 in any simple factor of .
These are equivalent notions.
Under the equivalence, and if for some , then can be written .
3 to 2: Let , then is a 1-eigenspace of and is compact. We have with eigenvalues .
Using to translate to all of yields and almost complex structure.
makes a complex manifold. Then there exists a -invariant symmetric and definite bilinear form on , so there exists -invariant riemannian metric which commtues with , and so is Hermitian. is , and so is noncompact.
(2) to (3): is adjoint and semisimple, so is for some . For , we get and , so is multiplication by on , because is an isolated fixed-point.
In fact, for any , there exists a unique isometry of such that on is mult by .
The uniqueness means that is a homomorphism, it algebraizes to over .
. Now, have eigenvalues .
5.3. Cartan’s Classification of irreducible HSD’s
Let be an irreducible HSD, a connected simple -alg group, a maximal algebraic torus over . where , . Over , defines a cocharacter which we may conjugate with any factor from and have for all .
Thus, must act through , and so or 1 for each . Then, is a miniscule coweight.
Therefore, for a unique simple root and for this , .
So we have a 1-1 correspondence between irreducible HSD and special nodes on a connected Dynkin diagram, and we can even say how many there are:
For , we get for , for , corresponds to K3 surfaces, for we get , which are the Siegel upper half planes.
6. Brosnan 4
6.1. Hodge Classes and Normal Functions
Lefschetz realized that you can start with a Hodge class where is a smooth projective algebraic surface, and if you assume that is primitive, meaning that for such that is smooth, then there is a normal function associated to in where is teh smooth locus of .
Let be a smooth projective complex variety. Then Deligne sets in degrees to .
Note that is just , and so gives the exponential sequence.
Thus, , and .
If is a smooth projective variety, then is an exact sequence .
Strictly, we’ve been working with hypercohomology, but we’re not going to worry about distinguishing it.
Suppose that is a morphism with smooth projective and irreducible, and the smooth locus. Write to be the primary classes.
Given , we can find a lift of ot . Then for every we have .
So for every , we get a map and so we write .
The association defines a normal function .
The dependence on is pretty simple: any other lift of will differ from by an element of .
We have a map .
So we have a triangle and so we jsut need to know . In fact, we have , and by degeneration of Hodge to deRham spectral sequence, we know that .
So , and therefor .
So we have .
6.2. A construction of Green-Griffiths
If is a smooth projective variety with and fixed very ample line bundle, from this let’s construct a sequence of normal functions on a sequence of spaces.
First, we construct the spaces. For each , set is the complete linear system on .
Now, write . Now for each we have an incidence variety and this is smooth over some dense open . Set .
By definition, if , and the projection, then , and so we get a map .
Now let , the set of singular hyperplane sections. Then Green-Griffiths noticed that teh singular of the normal function associated to a primitive class is related to the restriction of to the divisor