I gave my talk today. It seems to have gone over well, though I’m not entirely happy with it…but then, as with many people, I’m my own harshest critic. Here’s the notes from other talks, the notes from mine will follow in a separate post.
1. Doran – Normal forms for lattice polarized K3 surfaces and Siegel modular forms
Jont with Adrian Clingher
arXiv 1004.3335 and 1004.3503
For an elliptic curve, we can put any cubic into Weierstrass normal form . This gives a point for each one, and they are smooth away from .
They’re not unique, though, but the point in classes them up to isomorphism, so we have a coarse moduli space.
Why did we pick this form? Let be a toric Fano surface built from the normal fan to a reflexive polytope.
Now, a K3 surface has trivial and all are diffeomorphic. The intersection pairing on gives it a lattice structure .
The Neron-Severi group is and is spanned by the classes of algebraic curves.
We have two 19 dimensional families come from the 2-polarized K3’s which are double covers of branched along a smooth sextic and 4-polarized K3’s which are smooth quartic surfaces in .
Now, rather than requiring that every K3 has a single polarization, assume that they have polarizing lattices, to get a new moduli problem:
Let be an even nondegenerate lattice of signiture for , then an -polarized K3 is where is a K3 and is an embedding. This gives a dimesnsional moduli space of -polarized K3 surfaces.
Now, for corresponds to modular curves, dropping the last term gives modular surface, taking gives a modular threefold (which will be our focus) and gives a modular 4fold.
Consider the singular (ADE-singularities) quartice hypersurface given by , and call its minimal resolution. If or , then is a K3 surface with a canonical N-polarization. Conversely, given an N-polarization on a K3, there exist with or such that is the K3 we started with.
There is a coarse moduli space for N-polarized K3 surfaces and the inverse period map is given by with , , and .
A comment on the Siegel modular forms: We want to mention Igusa’s Theorem: for , we have for and for the cusp forms etc, they have some expressions in terms of the modular forms.
The graded ring of Siegel modular forms of degree is generated by and is isomorphic to a polynomial ring with , a degree 70 polynomial.
Now, we have all of these generators except for . Where does it come in? It turns out that , where are equations for the modular varieties we’ve obtained.
The Nikulin (symplectic) involution on a K3 is an analytic involution such that for any homorphic 2-form on .
- Fix point locus of consists of 8 distinct points
- the surface obtained by minimal resolution of the quotient is a K3
- There is a degree 2 rational map with branch locus given by eight disjoint rational curves.
- There is a pushforward morphism the orthogonal component of even curves in .
Shioda-?? Structure is a Nik inv, such that for a Kummer surface, the resolution of , the morphism induces a Hodge isomoetry between lattices of transcendental cocycles and .
A van Geemen-??? involution is an for which there exists a triple such that is an elliptic fibration on , are disjoint sections of , has order 2 in and is the involution obtained by extending the fiberwise translation by in the smooth fibers of , using the group structure with ideneity section given by .
vG-S inv is fiberwise 2-isogeny relating to . So we get . To make things symmetric, we’ll still need and inducing a vG-S structure on , and these exist.
Now, let be -polarized quartic surface, then we have , which cannot be extended to a polarization by , then can be extended to or it cannot be. We call the first case special and the second nonspecial.
If is a double cover of branched over a configuration of six lines, no three concurrent. Then the special configuration is when the six are tangent to a fixed conic, and this gives a Kummer surface, the non-special (nonKummer) is when there is no such smooth conic.
2. Usui – Neron Models in log mixed Hodge theory by weak fans
3. Carlson – Further speculation and progress on Hodge theory for cubic surfaces
Joint work with Domingo Toleda.
First, we’ll recall some facts about cubic curves.
Topologically, they’re tori, and can be written as where is a cubic. They have a presentation as where where is a number with positive imaginary part, and it is the ratio of the periods of the holomorphic 1-form.
Now, some cubic curves have extra automorphisms. For instance, if is a cube root of unity, then this curve has an automorphism of order 6, and if , then order 4.
Other elliptic curves may have interesting automorphisms, however. Generically, though, , but for CM curves, it is .
We have , and this is where , then if , and and . Now, we get a representation .
Also, we have embeddings , and we can take a diagonal matrix with entries which lies in . So in the case of , we have a rank 2 torus over our number field.
Now, onto cubic surfaces.
We have the algebraic classes.
A useful trick is that if is a cuibc surface, then we tcan take a cubic threefold and we get maps from the surface, to the cyclic (it’s a 3-to-1 cover of , so there’s a deck transformation, take generator to by ) cubic threefold, to the intermediate Jacobian of the cubic threefold.
Usually we talk about period matrices, but it’s also useful to look at period vectors. We can write the cubic threefold as and acts by .
Now, acts on , and we can split it into a direct sum of eigenspaces and . We can break these up into the Hodge decomposition, and we find that is one dimensional, spanned by .
Now, by using on , we have a -module structure, and we have .
Now, we set to be the vector with components , and we can show that , and so if we divide by to get , we have , so is the four-dimensional unit ball.
Now, let where is . Then the set of cubic sufraces maps to and these are both four dimensional, and in fact, it is an isomorphism.
The theorem needed for this is the Clemens-Griffiths Torelli Theorem for cubic threefolds.
Question: Are there special periods of cubic surfaces?
If , then is isogenous to
Proof: Cayley cubic surface is given by , and so a basis for the vanishing cycles are . Now we define to be the matrix
(I didn’t really follow this part)
4. Charles – Remarks on the Lefschetz standard conjecture and hyperkähler varieties
Let be a smooth projective variety of dimension .
by cupping with is an isomorphism.
There exists such that is the inverse of .
This conjecture is due to Grothendieck, and is called the Lefschetz standard conjecture in degree .
Above, stands for the Betti cohomology over a field , where is or .
And so, we construct a nice category of pure motives, as it implies the other standard conjectures, it implies the Hodge conjecture for Abelian varieties, the Künneth components of are algebraic, and that numerical and homological equivalence are the same.
Now, we know the Lefschetz Standard conjecture is ok in degree 1, which means it’s good for curves and surfaces, it’s true for abelian varieties, for varieties with cohomology spanned by algebraic cycles. These imply that we know it for varieties coming from these things, products, hyperplane sections, etc, and moduli spaces of sheaves on K3s and Abelian surfaces.
Now, we’re looking for , and this gives us a family of cycles on which are paramterized by itself.
Let be a smooth projective variety and of codimension .
Now, , which maps to , and further to .
The Lefschetz conjecture is true for in degree 2 iff there exists and as above, and such that is surjective.
Remark: We can ask for a formal Lefschetz conjecture.
Same result if we have on vector bundle such that is surjective.
Now, we want to look at hyperkähler varieties.
Let be hyperkähler, irreducible and projective, and on such that is hyperholomorphic, stable and non-rigid. Then satisfies the Lefschetz standard conjecture in degree 2.
If is a family of hyperkähler varieties and is a K3, the Hilbert scheme of points. This has a dimensional smooth family of deformations and a general projective one does not come from a K3. surface. Same with generalized Kummers
The Lefschetz conjecture is true for projective formation os .
Let be smooth projective. We recall a famous lemma:
Lemma 1 The Lefschetz conjecture is true in degree for if and only if there exists such that is an isomorphism.
This follows essentially from Cayley-Hamilton.
The Lefschetz conjecture is true in degree for iff there exists a smooth projective variety satisfying the Lefschetz conjecture in degree and such that, if , we have is surjective.
Proof: We would like for to be surjective.
Assume that there exists such that It means that for any on , we have , but , and the right isn’t zero.
Now, we use the Lefschetz decomposition and change the signs, for this, we need Lefschetz on .
The most interesting part is the component , and if this is surjective and , then we get everything but . Thus, for , we only need the surjectivity of this map.
By Serre duality, it corresponds to an element of , which is .
Lemma: This is .
Using chern classes, we get the vector bundle formulation. We need on such that is surjective.
Now, let be irred. hyperkähler. This is equivalent to irreducible holomorphic symplectic. Now, assume is simply-connected and is spanned by the symplectic for.
Given Kähler, this is equivalent to having complex structures with and (etc)
Now a vector bundle on is hyperholomroophic if it has a connection compatible with .
Fact: A given hyperkähler comes with a family of deformation parameterized by by with . Hyperholomorphic bundles deform along this .
Verbitsky shows that if is stable, then the reduced subscheme of the moduli space of def of has a compatible hyperkähler structure. This implies that the form we’re looking at is symplectic.
5. Maxim – Characteristic classes of complex hypersurfaces
6. Kerr – Mumford-Tate groups and the classification of Hodge structures
When is a given -algebraic group the Mumford-Tate group of some polarized Hodge structure?
Let be a vector space over and . Fix a nondegenerate -sym bilinear form, and the vector of with , and , and let the period domain for Hodge structures on polarized by with hodge numbers .
(Talk went very fast, couldn’t take notes.)
7. Siegel – The Schottky Problem
My talk, see separate notes.
8. Dalakov – Deformations of the Hitchin section and DGLA’s