## ICTP Day 14 – Doomsday

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

arXiv 1004.3335 and 1004.3503

For an elliptic curve, we can put any cubic into Weierstrass normal form ${Y^2Z-4X^3+g_2 XZ^2+g_3Z^3=0}$. This gives a point ${(g_2,g_3)\in \mathbb{C}^2}$ for each one, and they are smooth away from ${g_2^3-27g_3^2=0}$.

They’re not unique, though, but the point ${(g_2:g_3)}$ in ${WP(2,3)}$ classes them up to isomorphism, so we have a coarse moduli space.

Why did we pick this form? Let ${\mathbb{P}_\Delta}$ be a toric Fano surface built from the normal fan to a reflexive polytope.

Now, a K3 surface ${X/\mathbb{C}}$ has trivial ${K_X}$ and all are diffeomorphic. The intersection pairing on ${H_2(X,\mathbb{Z})}$ gives it a lattice structure ${H\oplus H\oplus H\oplus E_8\oplus E_8=(L,\langle\rangle)}$.

The Neron-Severi group is ${H^{1,1}(X,\mathbb{Z})}$ 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 ${\mathbb{P}^2}$ branched along a smooth sextic and 4-polarized K3’s which are smooth quartic surfaces in ${\mathbb{P}^3}$.

Now, rather than requiring that every K3 has a single polarization, assume that they have polarizing lattices, to get a new moduli problem:

Let ${M}$ be an even nondegenerate lattice of signiture ${(1,t)}$ for ${t\leq 19}$, then an ${M}$-polarized K3 is ${(X,\iota)}$ where ${X}$ is a K3 and ${\iota:M\rightarrow NS(X)\subset L}$ is an embedding. This gives a ${(19-t)}$ dimesnsional moduli space of ${M}$-polarized K3 surfaces.

Now, for ${H\oplus E_8\oplus E_8\oplus \langle z_0\rangle\subset L}$ corresponds to modular curves, dropping the last term gives modular surface, taking ${N=H\oplus E_8\oplus E_7}$ gives a modular threefold (which will be our focus) and ${H\oplus E_7\oplus E_7}$ gives a modular 4fold.

Consider the singular (ADE-singularities) quartice hypersurface ${Q(\alpha,\beta,\gamma,\delta)\subset \mathbb{C}\mathbb{P}^3}$ given by ${Y^2ZW-4X^3Z+3\alpha XZW^2+\beta ZW^3+\gamma XZ^2W-\frac{1}{2}(\delta Z^2W^2+W^4)=0}$, and call ${X(\alpha,\beta,\gamma,\delta)}$ its minimal resolution. If ${\gamma\neq 0}$ or ${\delta\neq 0}$, then ${X(\alpha,\beta,\gamma,\delta)}$ is a K3 surface with a canonical N-polarization. Conversely, given an N-polarization on a K3, there exist ${\alpha,\beta,\gamma,\delta}$ with ${\delta\neq0}$ or ${\gamma\neq0}$ such that ${X(\alpha,\beta,\gamma,\delta)}$ is the K3 we started with.

There is a coarse moduli space for N-polarized K3 surfaces ${\mathscr{M}^{N-pol}_{K3}=\{(\alpha:\beta:\gamma:\delta)|WP(2,3,5,6)|\gamma\neq 0\mbox{ or }\delta\neq 0\}}$ and the inverse period map is given by ${per^{-1}=(\alpha:\beta:\gamma:\delta)}$ with ${\alpha=\mathscr{E}_4}$, ${\beta=\mathscr{E}_6}$, ${\gamma=2^{12}3^{5}\mathscr{C}_{10}}$ and ${\delta=2^{12}3^6 \mathscr{C}_{12}}$.

A comment on the Siegel modular forms: We want to mention Igusa’s Theorem: for ${\kappa\in \mathbb{H}_1}$, we have ${\mathscr{E}_{2t}(\kappa)=\sum_{(C,D)} \det(C\kappa +D)^{-2t}}$ for ${t>1}$ and for the cusp forms ${\mathscr{C}_{10}, \mathscr{C}_{12}}$ etc, they have some expressions in terms of the modular forms.

The graded ring ${A(Sp(4,\mathbb{Z}),\mathbb{C})}$ of Siegel modular forms of degree ${\geq 4}$ is generated by ${\mathscr{E}_4,\mathscr{E}_6,\mathscr{C}_{10},\mathscr{C}_{12},\mathscr{C}_{35}}$ and is isomorphic to a polynomial ring with ${\mathscr{C}_{35}^2=Poly_{70}}$, a degree 70 polynomial.

Now, we have all of these generators except for ${\mathscr{C}_{35}}$. Where does it come in? It turns out that ${\mathscr{C}_{35}=\mathscr{D}_4\mathscr{D}_1}$, where ${\mathscr{D}_i}$ are equations for the modular varieties we’ve obtained.

The Nikulin (symplectic) involution on a K3 ${X}$ is an analytic involution ${\alpha:X\rightarrow X}$ such that ${\alpha^*\omega=\omega}$ for any homorphic 2-form ${\omega}$ on ${X}$.

\mbox{}

1. Fix point locus of ${\alpha}$ consists of 8 distinct points
2. the surface ${Y}$ obtained by minimal resolution of the quotient is a K3
3. There is a degree 2 rational map ${p_\alpha:\dashrightarrow Y}$ with branch locus given by eight disjoint rational curves.
4. There is a pushforward morphism ${(p_\alpha)_*:H^2(X,\mathbb{Z})\rightarrow H_Y}$ the orthogonal component of even curves in ${H^2(Y,\mathbb{Z})}$.

Shioda-?? Structure is a Nik inv, such that for ${Y}$ a Kummer surface, the resolution of ${A/-1}$, the morphism ${(p_\alpha)_*}$ induces a Hodge isomoetry between lattices of transcendental cocycles ${T_X(Z)}$ and ${T_Y}$.

A van Geemen-??? involution is an ${\alpha_X: X\rightarrow X}$ for which there exists a triple ${(\phi_X,S_1,S_2)}$ such that ${\phi_X:X\rightarrow \mathbb{P}^1}$ is an elliptic fibration on ${X}$, ${S_1,S_2}$ are disjoint sections of ${\phi_X}$, ${S_2}$ has order 2 in ${MW(\phi_X,S_1)}$ and ${\alpha_X}$ is the involution obtained by extending the fiberwise translation by ${S_2}$ in the smooth fibers of ${\phi_X}$, using the group structure with ideneity section given by ${S_1}$.

vG-S inv is fiberwise 2-isogeny relating ${X}$ to ${Y}$. So we get ${\phi_Y:Y\rightarrow \mathbb{P}^1}$. To make things symmetric, we’ll still need ${S_1'}$ and ${S_2'}$ inducing a vG-S structure on ${Y}$, and these exist.

Now, let ${X}$ be ${H\oplus E_7\oplus E_7}$-polarized quartic surface, then we have ${\iota:H\oplus E_7\oplus E_7\rightarrow NS(X)}$, which cannot be extended to a polarization by ${H\oplus E_8\oplus E_8}$, then ${\iota}$ can be extended to ${N}$ or it cannot be. We call the first case special and the second nonspecial.

If ${Y}$ is a double cover of ${\mathbb{C}\mathbb{P}^2}$ 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

Slides

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 ${y^2=P(x)}$ where ${P(x)}$ is a cubic. They have a presentation as ${\mathbb{C}/L}$ where ${L=\mathbb{Z}\{1,\tau\}}$ where ${\tau}$ 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 ${\tau=\omega}$ is a cube root of unity, then this curve has an automorphism of order 6, and if ${\tau=i}$, then order 4.

Other elliptic curves may have interesting automorphisms, however. Generically, though, ${\mathrm{End}(E)\otimes\mathbb{Q}=\mathbb{Q}}$, but for CM curves, it is ${\mathbb{Q}(\sqrt{-d})}$.

We have ${MT=SL(2)}$, and this is ${T_K}$ where ${K=\mathbb{Q}(\sqrt{-d})}$, then if ${\lambda\in T_K(R)=(K\otimes \mathbb{R})^\vee}$, and ${T_k(\mathbb{Q})=K^*}$ and ${K=\mathbb{Q}(0)}$. Now, we get a representation ${\rho:T_k(\mathbb{Q})\rightarrow GL_d(\mathbb{Q})}$.

Also, we have embeddings ${\tau_i:K\rightarrow\mathbb{C}}$, and we can take a diagonal matrix with entries ${\tau_i(\lambda)}$ which lies in ${GL_d(K)}$. So in the case of ${T_K}$, we have a rank 2 torus over our number field.

Now, onto cubic surfaces.

We have ${H^2(X)=H^{1,1}(X)}$ the algebraic classes.

A useful trick is that if ${X\subset\mathbb{P}^3}$ is a cuibc surface, then we tcan take ${Y}$ a cubic threefold and we get maps from the surface, to the cyclic (it’s a 3-to-1 cover of ${\mathbb{P}^3}$, so there’s a deck transformation, take generator to by ${\sigma}$) 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 ${x_4^3+F(x_0,\ldots,x_3)=0}$ and ${\sigma}$ acts by ${\omega x_4}$.

Now, ${\sigma}$ acts on ${H^3(Y,\mathbb{C})}$, and we can split it into a direct sum of eigenspaces ${H^3(Y)_\omega}$ and ${H^3(Y)_{\bar{\omega}}}$. We can break these up into the Hodge decomposition, and we find that ${H^{2,1}_{\omega}}$ is one dimensional, spanned by ${\Phi}$.

Now, by using ${\sigma}$ on ${H_3(Y,\mathbb{Z})}$, we have a ${\mathbb{Z}[\omega]}$-module structure, and we have ${2h(x,y)=Q(\sigma x,\bar{y})-\omega Q(x,\bar{y})}$.

Now, we set ${P}$ to be the vector with components ${\int_{\gamma_i} \Phi=P_i}$, and we can show that ${-|P_0|^2+\ldots+|P_4|^2<0}$, and so if we divide by ${P_0}$ to get ${Z_i}$, we have ${|Z|^2<1}$, so ${Z(X)}$ is the four-dimensional unit ball.

Now, let ${\Gamma=U(h,\mathscr{O})}$ where ${\mathscr{O}}$ is ${\mathbb{Z}[\omega]}$. Then the set of cubic sufraces maps to ${B^4/\Gamma}$ 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? ${Z\in \mathbb{Q}(\omega)^4=K^4}$

If ${Z\in \mathbb{Q}(\omega)^4}$, then ${J(X)}$ is isogenous to ${E^5_\omega}$

Proof: Cayley cubic surface is given by ${\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=0}$, and so a basis for the vanishing cycles are ${e_1,\ldots,e_4}$. Now we define ${\hat{P}}$ to be the matrix ${\left(\begin{array}{cccc}1&z&\omega &\bar{\omega}z\\ i_Z & I& \bar{\omega}^t Z& \omega I\end{array}\right)}$

(I didn’t really follow this part) $\Box$

4. Charles – Remarks on the Lefschetz standard conjecture and hyperkähler varieties

Let ${X/\mathbb{C}}$ be a smooth projective variety of dimension ${n}$.

${L^{n-k}:H^k(X)\rightarrow H^{2n-k}(X)}$ by cupping with ${\omega^{n-k}}$ is an isomorphism.

There exists ${Z\in CH^n(X\times X)}$ such that ${[Z]_*:H^{2n-k}(X)\rightarrow H^k(X)}$ is the inverse of ${L^{n-k}}$.

This conjecture is due to Grothendieck, and is called the Lefschetz standard conjecture in degree ${k}$.

Above, ${H^k(X)}$ stands for the Betti cohomology over a field ${K}$, where ${K}$ is ${\mathbb{Q},\mathbb{R}}$ or ${\mathbb{C}}$.

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 ${\Delta\subset X\times X}$ 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 ${Z\subset X\times X}$, and this gives us a family of cycles on ${X}$ which are paramterized by ${X}$ itself.

Let ${S}$ be a smooth projective variety and ${Z\subset X\times S}$ of codimension ${k}$.

Now, ${[Z]\in H^k(X\times S, \Omega^k_{X\times S})}$, which maps to ${H^0(S,R^k\pi_*\Omega^k_{X\times S})}$, and further to ${H^0(S,\Omega^k_S)\otimes H^k(X,\mathscr{O}_X)=\hom(\bigwedge^k T_S,\mathscr{O}_S\times H^k(X,\mathscr{O}_X)}$.

The Lefschetz conjecture is true for ${X}$ in degree 2 iff there exists ${S}$ and ${Z}$ as above, and ${s\in S(\mathbb{C})}$ such that ${\phi_{Z,s}:\bigwedge^2 T_{S,s}\rightarrow H^2(X,\mathscr{O}_X)}$ is surjective.

Remark: We can ask for a formal Lefschetz conjecture.

Same result if we have ${\mathscr{E}}$ on ${X\times S}$ vector bundle such that ${\phi_{S,s}:\bigwedge^2 T_{S,s}\stackrel{KS}{\rightarrow} \bigwedge^2 \mathrm{Ext}^1(\mathscr{E}_s,\mathscr{E}_s)\rightarrow \mathrm{Ext}^2(\mathscr{E}_s,\mathscr{E}_s)\stackrel{tr}{\rightarrow} H^2(X,\mathscr{O}_X)}$ is surjective.

Now, we want to look at hyperkähler varieties.

Let ${X}$ be hyperkähler, irreducible and projective, and ${\mathscr{E}}$ on ${X}$ such that ${\mathscr{E}}$ is hyperholomorphic, stable and non-rigid. Then ${X}$ satisfies the Lefschetz standard conjecture in degree 2.

If ${T}$ is a family of hyperkähler varieties and ${S}$ is a K3, ${S^{[n]}}$ the Hilbert scheme of points. This has a ${22}$ 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 ${S^{[n]}}$.

Let ${X/\mathbb{C}}$ be smooth projective. We recall a famous lemma:

Lemma 1 The Lefschetz conjecture is true in degree ${k}$ for ${X}$ if and only if there exists ${Z\in CH^k(X\times X)}$ such that ${[Z]_*:H^{2n-k}(X)\rightarrow H^k(X)}$ is an isomorphism.

This follows essentially from Cayley-Hamilton.

The Lefschetz conjecture is true in degree ${k}$ for ${X}$ iff there exists ${S}$ a smooth projective variety satisfying the Lefschetz conjecture in degree ${\leq k-2}$ and ${Z\in CH^k(X\times S)}$ such that, if ${\ell=\dim S}$, we have ${[Z]_*:H^{2\ell-k}(S)\rightarrow H^k(X)}$ is surjective.

Proof: We would like for ${[Z]_*\circ L^{\ell-k}\circ [Z]^*:H^{2n-k}(X)\rightarrow H^k(X)}$ to be surjective.

Assume that there exists ${\alpha\in H^{2n-k}(X)}$ such that ${[Z]_* L^{\ell-k} [Z]^*\alpha=0}$ It means that for any ${\beta}$ on ${H^{2n-k}(X)}$, we have ${\langle\beta,[Z]_*L^{\ell-k}[Z]^*\alpha\rangle=0}$, but ${\langle [Z]^*\beta,L^{\ell-k}[Z]^*\alpha\rangle=0}$, and the right isn’t zero.

Now, we use the Lefschetz decomposition and change the signs, for this, we need Lefschetz on ${S}$.

The most interesting part is the component ${H^\ell(S,\Omega^{\ell-k}_S)\rightarrow H^k(X,\mathscr{O}_X)}$, and if this is surjective and ${k<\ell}$, then we get everything but ${NS(X)}$. Thus, for ${k=2}$, we only need the surjectivity of this map.

By Serre duality, it corresponds to an element of ${H^0(S\Omega^k_S)\otimes H^k(X,\mathscr{O}_S)}$, which is ${\hom(\bigwedge^k T_S,\mathscr{O}_S\otimes H^k(X,\mathscr{O}_X))}$.

Lemma: This is ${\phi_Z}$.

Using chern classes, we get the vector bundle formulation. We need ${\mathscr{E}}$ on ${X\times S}$ such that ${\bigwedge^2T_S\stackrel{KS}{\rightarrow} \bigwedge^2 \mathrm{Ext}^1(\mathscr{E}_s,\mathscr{E}_s)\rightarrow \mathrm{Ext}^2(\mathscr{E}_s,\mathscr{E}_s)\rightarrow H^2(X,\mathscr{O}_X)}$ is surjective. $\Box$

Now, let ${X/\mathbb{C}}$ be irred. hyperkähler. This is equivalent to irreducible holomorphic symplectic. Now, assume ${X}$ is simply-connected and ${H^0(X,\Omega^2_X)}$ is spanned by the symplectic for.

Given ${X}$ Kähler, this is equivalent to having ${I,J,K}$ complex structures with ${IJ=-JI=K}$ and ${JK=I}$ (etc)

Now ${\mathscr{E}}$ a vector bundle on ${X}$ is hyperholomroophic if it has a connection compatible with ${I,J,K}$.

Fact: A given ${X}$ hyperkähler comes with a family of deformation parameterized by ${\mathbb{P}^1}$ by ${aI+bJ+cK}$ with ${a^2+b^2+c^2=1}$. Hyperholomorphic bundles deform along this ${\mathbb{P}^1}$.

Verbitsky shows that if ${\mathscr{E}}$ is stable, then the reduced subscheme of the moduli space of def of ${\mathscr{E}}$ has a compatible hyperkähler structure. This implies that the form we’re looking at is symplectic.

5. Maxim – Characteristic classes of complex hypersurfaces

On Slides

6. Kerr – Mumford-Tate groups and the classification of Hodge structures

When is a given ${\mathbb{Q}}$-algebraic group the Mumford-Tate group of some polarized Hodge structure?

Let ${V}$ be a vector space over ${\mathbb{Q}}$ and ${n\in \mathbb{Z}}$. Fix ${Q:V\times V\rightarrow \mathbb{Q}}$ a nondegenerate ${(-1)^n}$-sym bilinear form, and ${\underline{h}}$ the vector of ${h^{p,q}}$ with ${p+q=n}$, ${\sum h^{p,q}=\dim V}$ and ${h^{p,q}=h^{q,p}}$, and let ${D_{\underline{h}}}$ the period domain for Hodge structures ${\phi}$ on ${V}$ polarized by ${Q}$ with hodge numbers ${\underline{h}}$.

(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

Slides