Today we started the third part of the summer school: Algebraic Cycles, Arithmetic aspects of Hodge Structures-Mumford-Tate Groups. There was actually quite a bit of review today, which surprised me, and instead of taking notes on the definition of Hodge structures again, I just noted that that’s what we did, and moved on.
1. Carlson 1 – Period Domains
Period domains are parameter spaces for marked Hodge structures. We call the period space, which is a parameter space of isomorphism classes of Hodge structures.
Let be a morphism of varieties, the fiber over and the subset of where the fibers are singular. Then the base change to is a smooth family. This induces a period map by with its Hodge structure. Really we lift to the universal cover, which maps into and associates to its marked Hodge structure.
(Review of what a Hodge structure is, and then Primitive Cohomology)
So what is a marking on a Hodge structure? Take a Hodge structure and with polarization . The lattice is isomorphic to for some . A marking is an isomorphism and on such that is an isometry.
Now, we set for curves, that is, rank 1 Hodge structures, and the more general definition is similar.
Now let be the isometry group of , and it is a discontinuous group, and is an analytic space.
2. Murre 1 – Chow Groups
Conventions: is an algebraically closed field, are varieties over , which are projetive (at worst, quasi-projective), irreducible and smooth.
2.1. Algebraic Cycles
Let be of dimension . Let and . Then be the group of algebraic cycles on of dimension (codimension ).
- is the set of Weil divisors
- is teh group of zero cycles, which are formal sums of points.
- on a threefold is the free abelian group on the curves on .
2.2. Operators on Cycles
There are three standard operations. The first is Cartesian product, if and , then .
The second is pushdown of cycles. If is a morphism and , then is defined as follows: take . If is the set theoretic image of an irreducible subvariety, then . If , then . If , then we have a finite extension of fields, and we set , where is always taken set-theoretically.
The third operation is the intersection product of two cycles, but it is not always defined. Let be subvarieties of smooth, of codimension and . Then where the are irreducible subvarieties of codimension at most . The intersection is proper if the codimension is exactly . We define the intersection multiplicity at to be
If is a surface and are curves and a point on , then we define the multiplicity at to be where are defining local equations for and . But this isn’t the right definition if .
In general, we take where .
And so, we define to be the intersection product if all the intersections are “good”. This then extends by linearity to .
There are other operations: let and , then the pullback.
We also have operation by correspondences, in particular if , then we have that is the image under the projection of .
Let and be varieties of dimension and and let be a correspondence from to . If , then is just the pushforward along the projection to of in .
2.3. Good=adequate equivalence relations on cycles
Samuel in 1956 said that on the groups of algebraic cycles is an adequate relation if:
- The set of cycles equivalent to zero should be a subgroup of .
- If are equivalent and , such that and are defined, then they are equivalent.
- If and , then there exists equivalent to such that is defined. (This is called the moving lemma, which motivated it)
If is good, then , the set of equivalence classes of this equivalence relation, are defined and is a commutative ring with respect to the intersection product, and are defined when is proper.
Some commonly used adequate equivalence relations are
- Rational equivalence (Chow, Samuel 1956)
- Algebraic equivalence (Weil 1952)
- Homological equivalence
- Numerical equivalence
2.4. Rational Equivalence
This is a generalization of linear equivalence.
Linear equivalence is a relation on . Here, for every , we have for smooth. If is not smooth, need to use , which can be reviewed in Hartshorne.
For , and , we say that is rationally equivalent to zero if where denotes the subgroup generated by cycles of the form where for .
Equivalently, there exists a finite collection of codimension subvarieties of .
3. Brosnan 1 – Normal Functions
The motivation is that Lefschetz used normal functions to prove the theorem, so maybe understanding normal functions will help prove the Hodge conjecture. The first person to explicitly define admissable normal functions seems to have been M. Saito in JAG and studied their zero loci and related them to the Hodge conjecture.
Let be a complex manifold and a variation of pure Hodge structure of negative weight on , a Zariski open subset of . Let be the group of admissable normal functions on . For , set be the zero locus of . Then the closure of in is a complex analytic subvariety of .
If is algebraic and projective, then is an algebraic subvariety of .
Remarks: There are at least two proofs of the above. One by Brosnan and Pearlstein, using information from the mixed orbit theorem of Kato, Nakayama and Usui. The other is by Schnell. He introduces an extension of the family of Griffiths intermediate Jacobians. Uses that to extend the normal functions and the idea of Kato-Nakayama-Usui to compactify is a more-or-less toroidal way.
There were some prior results by Saito in a JAG paper and by Brosnan-Pearlstein when . Also when the singularity of vanishes.
There are two basics: the first is Hodge structures. The category of pure Hodge structures of weight is by definition the category of pairs such that is finitely generated abelian group, are subspaces of such that is their direct sum and , and morphisms are maps preserivng the ‘s. We can replace finitely generated abelian group with -modules for a subgroup of , though are the only useful ones.
The category of MHS’s is an abelian category, and if is an algebraic variety over , then carries an MHS.
The second part is much more difficult than the first, adn involves geometry and a lot of homological algebra. The first part is just a masterpiece of linear algebra. The idea is to define , as in El Zein’s notes. These give a bigrading, and a morphism of MHS’s is a morphism that preserves the , and that’s enough to show that it’s an abelian category.
The category of pure Hodge structures is essentially semi-simple. If is a smooth projective variety then is a direct sum of irreducible Hodge structures. The category of MHS is not, there are nontrivial extension.
Take and set . Then is a nontrivial extension of by .
In fact, we can explain the extension geometrically we have which is naturally .
To see that the extension is nontrivial, really need to calculate it, but to calculate it, we need to find where it goes.
Let be pure Hodge structures with of lower weight than and torsion free. Then .
There, is a pure Hodge structure of weight where are the homomorphisms .
Proof: We’ll define a map by taking an extension . We can always find of finitely generated abelian groups splitting the extension. On the other hand, maps of mixed Hodge structure are strict with respect to the Hodge filtration, so , and from this it follows that we can find a map such that is in the right place in the filtration.
4. Carlson 2
So our set up is that is a period domain, it’s the set of ‘s, and we have a marked polarized Hodge structure . We really want to describe the period domain in terms of the Hodge filtration.
In the weight 1 case, , and it needs to satisfy so it is contained in a proper subvariety, but then there’s also where , so there’s some open conditions. So .
In the weight 2 case, we first note that for polarized Hodge structures, then determines . So we have where and .
Let’s look at elliptic curves. Set where is a cubic with distinct roots. For example, , so long as we get an elliptic curve, which is a nonsingular Riemann surface of genus 1.
The unique (up to scaling) abelian differential is .
Show that is holomorphic at .
So, we can write . We define a marking , picking out a symplectic basis for the cohomology. Take the basis to be , and the dual basis .
We know that , and that and so , which means that , and so we get that .
Consequence of Riemann Bilinear relation is that . So we can rescale so that the -period is 1. So the period matrix because . What can we say about ? The relation is now , so we have that .
So we now know that .
4.1. Period Map
Let a morphism and of maximal rank on . The smooth part of the map is locally differentiably trivial. The period map is the map and also the map it induces .
Look at the family of elliptic curves. We claim that the period is defined and holomorphic on an open set away from in . Take our marking to pick the basis with intersection . Then and .
These are integrals with fixed domains of integration, so we can take the derivative of the inside to check holomorphicisty, and it’s clear that the -period is holomorphic, and so is the -period for the same reasons.
So the period map is . So we have the period map. And then there’s .
Show that .
So, we get a map .
Let be a fundamental region for . Almost al of is given by . No two points of are equivalent under , so if , then there is no such that .
The space is an orbifold, not quite a manifold, it has a few special points.
Reference for Modular Forms: Serre’s Course in Arithmetic.
Set and say it has weight 4 and set of weight 6. These are modular forms, they’re not invariant under , but transform in a controlled way. Define a polynomial in them by and set . We call the discriminant.
If the elliptic curve is (and all complex elliptic curves can be put in this form by a change of variables) then if and only if the curve is singular.
The function gives a biholomorphic map .
5. Murre 2
5.1. Rational Equivalence
We have a cycle where and .
Lemma 1 The following conditions on are equivalent:
- is rationally equivalent to zero
- there exists and such that .
Proof: 1 implies 2 is easy. For the other direction, we use a theorem of Fulton, which says that if is proper and and , then .
Take the condition for . and . .
Rational equivalence is a “good” equivalence relation. is a subgroup, for the second condition, let such that and are defined for any given . Then take and . Because we can assume no horizontal components, it works.
The interesting part is the moving lemma. Let and choose finitely many . We want to find such that are defined.
Case 1: , a general projective transforming on . and are rationally equivalent, so we can make sure is defined.
Case 2: where is called the excess. Take a general linear space . Take the cone spanned by and . Then . , the excess intersection. Because is general, the excess is , and things work out.
5.2. Chow Groups
We define .
- is a commutative ring with identity.
- behaves functorially with respect to . More precisely:
- if is proper, then is
- if arbitrary, is
- gives is a homomorphism.
Let inclusions. Then is exact.
induces is an isomorphism for .
5.3. Algebraic equivalence
Let , we say that is algebraically equivalent to zero if there exists such that where and is a curve.
It’s easy to see that , because can be taken as .
5.4. Homological Equivalence
Fixing a good cohomology theory, for instance, the usual ones, but also etale cohomology will work. Then we get the usual intersection theory in cohomology.
We require that there exist a cycle map such that the intersection product is compatible with the cohomological cup product. Set to be the homologically trivial cycles. This is also a good equivalence relation.
, by Matsusaka in 1956, it was shown for divisors. It is not true in general, proved in 1969.
5.5. Numerical Equivalence
Let be a -dimensional projective variety, and such that is defined, and it’s just a number of points, a zero cycle. We say that is numerically equivalent to zero if for all where it is defined.
Is ? It is known for divisor, and it’s a conjecture for general .
For a variety over , the Hodge conjecture implies this one.
6. Brosnan 2
We’ll finish proving the theorem of Carlson.
So if we start with , because is torsion free, we can find a splitting because is strict with respect to , we can find a splitting preserving the Hodge filtration.
Using the fact that to decude that is strict with respect to teh Hodge filtration. Define . But depends on some choices: we could add any morphism to , and we could add any morphism to . Modding out by these choices determines the morphism we need.
The second step is to product a map backwards. Suppose that . Define , and define by . Define an extension by taking the underlying group to be , and . Then induces the identity on and therefore on and therefore is a Hodge structure. We need to check that .
Most important case of the theorem is if and is of negative weight, we set , this is a complex torus. To check this we need to check that is discrete in . In fact, .
In the exercise, we have . In that case, we have (all previous ‘s are )
So this is just , which is and by the exponential map, this is .
In general, is called the Griffiths intermediate Jacobian, and is generally not algebraic. However, when is polarized, then is an abelian variety: is the Jacobian of the curve.
If the weight is , then is a compact complex group. The reason is that we need to show that i sdiscrete. Look at . Ths is injective because is zero, both have the same dimension as real vector spaces, and so it is an isomorphism of real vector spaces. Since is discrete in , it is in as well.
Let be a smooth projective complex curve, and a divisor of degree 0. We get a long exact sequence of MHS as follows:
and the last two terms are and . The divisor then gives a map by sending to .
So gives an extension , which is just the Jacobian of , and the map is the Abel-Jacobi map .
6.1. Normal Functions
Let be a divisor on a surface where the are irreducible curves not contained in fibers of a map .
For each we get a curve as the fiber, and on a dense open , these curves are smooth.
Assume that for one and helce all , we have has degree . Then the above construction applied to the curves gives a section of the family whose fibers are the Jacobians of . For eahc point, we’ve got the Abel-Jacobi map, and this section is called a normal function.