Today was a half day, and we finished Carlson’s series. I think this afternoon I’ll clean some things up and maybe post notes from the finished series on my website, along with links from here. In any case, here are my notes:
1. Murre 4
We have an irreducible smooth variety. Look at the complex torus , the intermediate Jacobian of Griffiths. It won’t in general be an abelian variety.
We define to be the largest abelian subvariety of .
1.1. Abel-Jacobi Map
There exists a homomorphism which factors through .
an outlike of the construction, we note that and are dual, and we take , the universal cover of and , which are also dual. Now, for , we have where is a topological chain of dimension .
So take , and take the cohomology class of representing this , for does not depend on the choice of cohomology class, , and we can choose such that it contains at least ‘s. Then .
Now, take , and we have and , and . So we have a well defined element .
Let be a curve, then , and if we take , a basis for , then .
Remark: Suppose that . We claim that .
1.2. Algebraic Equivalence vs. Homological Equivalence
We look at . For divisors , we have equality for every algebraically closed by a theorem of Matsusaka in 1956.
There exists smooth over such that for certain the .
We define to be the Griffiths group, which is discrete.
There exist varieties such that for , and, in fact with .
There exist Chow varieties, given fixing the dimension and degree , then is the Chow variety.
Lefschetz hyperplane Theorem: with a hyperplane, is an isom for and is injective for .
In particular, a hypersurface section then for odd and and where for and .
We need the notion of a Lefschetz pencil: Let . Take hyperplanes in general position with respect to , then for is a pencil, and we call a Lefschetz pencil.
- For all except finitely many , is smooth
- for only are ordinary double points, where is the locus of singular sections.
The action of (the image of the monodromy representation) on is completely reducible.
Fact: if where the vanishing means it is in the kernel of .
Let and . Consider a Lefschetz pencil on with , and assume that , but . Let and set . Assume that for very general , the is algebraically equivalent to zero. Then is homologically equivalent to 0 on .
Indication of proof: Let , then step 1 is to look at , then to take a normal function , and step 2 is to show that is homologically zero on .
2. Carlson 5
Recall that the set of meromorphic forms with a pole at . By the adjoint of the tube map, we have a residue map to .
This map is holomorphic, because if we take , and for Horizontality, we take , and this is essentially Griffiths transversality for hypersurfaces.
Now, let be a period domain of filtrations. It’s an open subset of a closed subset of a product of Grassmannians . What are the tangent spaces of a Grassmannian?
Look at . For , we have that .
So in general we have that , and this lies in .
Now take where is or , and , then which is a HS of weight 0, inducing one on .
Let be a -module, these correspond to homogeneous vector bundles on by taking , corresponds to the Holomorphic tangent bundle of and is the horizontal part, and a family gives a family of Hodge structures, .
So then we have with fiber compact complex, and a symmetric space. Then we set and .
2.1. Curvature of
It is fairly well known that has holomorphic sectional curvature negative and bounded above by a negative constant. On more general, the sectional curvatures can be positive or negative: the positive ones correspond to for even, and for odd we get negative directions, and especially important is .
Distance Decreasing Principal: Let a holomorphic, horizontal, negatively curved and bounded away from zero manifold, then this map is distance decreasing. Specifically, .
If is a disk of radius , then .
A nontrivial variation of Hodge structures has at least 3 singularities.
For elliptic curves, assume not. then we have , this lifts to the universal cover the unit disc, and this gives an entire holomorphic function, which is a contradiction.
In generality, we look at and . Now, we have This is then for and in fact , and so , which is a contradiction.
Application 2: The eigenvalues of a monodromy transformation are roots of unity. Let map the unit disc to and . We’ll look at . Then . Now, , so the RHS is .
We have , which rewrites LHS as , and this is just when we write it as , and so we have , but we can use the homogeneous nature of our space to write .
So, the conjugacy class of has a limit point in (compact), os the eigenvalues of have absolute value , and are algebraic integers, so then a theorem of Kronecker that implies that they are roots of unity.
3. Griffiths 1 – Mumford-Tate Groups
Mumford-Tate groups are a rich class of groups and are the basic symmetry groups of Hodge structures. They capture both the and structures.
They sit at the intersection of algebraic geometry, representation theory and arithmetic geometry.
We’re going to focus on the higher weight case, because the classical case is much more well understood.
Some notation: a -vector space, its tensor product with , etc, the dual, a nondegenerate bilinear form which is alternating or symmetric, and as a rational algebraic group.
We’ll denote by the restriction of scalars.
This talk was given from the lecture notes on an overhead, and went too fast to take notes.