The Schottky Problem (ICTP)

These are my notes, and are only a rough approximation of the actual talk:

1. General Torelli and Schottky Problems

Take a class of smooth projective varieties with the same Hodge numbers. We can immediately ask two questions about the map from the set of these objects to the period domain:

1. Torelli: Is the map injective? If not, how does injectivity fail?
2. Schottky: What is the image? That is, which Hodge structures can be achieved?

We’ll look at a specific case, the one that’s usually the simplest: smooth curves of genus ${g}$.

In this case, the map associates to each curve its Jacobian: ${\mathcal{J}:\mathcal{M}_g\rightarrow\mathcal{A}_g}$. It is well known, in this case, that ${\mathcal{J}}$ is injective. However, determining the closure of the image, call it ${\mathcal{J}_g}$, is much more difficult.

2. Moduli and Maps

We start by defining some moduli spaces (we’ll actually need some compactifications, but we won’t worry about those details):

1. ${\mathcal{RM}_g=\{(C,\mu)|C\in\mathcal{M}_g,0\neq\mu\in J(C)\}=}$ ${\{(C,\tilde{C})|\tilde{C}\rightarrow C}\mbox{ etale double cover } {\}}$
2. ${\mathcal{RA}_g=\{(A,\mu)|A\in\mathcal{A}_g,0\neq\mu\in A\}}$

Also, we get maps ${\alpha:\mathcal{A}_g\rightarrow \mathbb{P}(U_g)}$ and ${\beta:\mathcal{RA}_g\rightarrow \mathbb{P}(U_{g-1})}$ by using the classical second order theta functions, which are related to symmetric divisors in the principal polarization, and ${U_g}$ is a vector space of dimension ${2^g}$. (It is actually a specific one built out of a representation of a Heisenberg group)

Finally, we define ${\mathcal{P}:\mathcal{RM}_g\rightarrow \mathcal{A}_{g-1}}$ to be the Prym map, which takes a double cover ${\tilde{C}\rightarrow C}$ of curves to ${\ker^0\mathrm{Nm}}$, which can be shown to be a ppav.

3. Schottky-Jung

In 1888, Schottky wrote down a modular form for genus 4, which he claimed vanished precisely on ${\mathcal{J}_4}$. Igusa announced a proof in 1968 and published it in 1981.

In 1909, in a joint paper with Jung, the Schottky-Jung relation was proved $\displaystyle \begin{array}{ccc}\mathcal{RM}_g & \stackrel{\mathcal{J}}{\to} & \mathcal{RA}_g \\ \mathcal{P} \downarrow & & \downarrow \beta \\ \mathcal{A}_{g-1} & \stackrel{\to}{\alpha} & \mathbb{P}(U_g)\end{array}$

In that paper, Schottky and Jung conjectured that if we set ${\mathcal{RS}_g=\beta^{-1}(\mathrm{im} \alpha)}$, and ${\mathcal{S}_g}$ the image of ${\mathcal{RS}_g}$ in ${\mathcal{A}_g}$, then ${\mathcal{S}_g=\mathcal{J}_g}$. This would allow us to actually write down equations for ${\mathcal{J}_g}$ in the natural coordinates on ${\mathcal{A}_g}$.

4. Non-Jacobians and big Schottky

It turns out that the SJ conjecture is false. There is a trick called the tetragonal construction which shows that the intermediate Jacobians of cubic threefolds are in ${\mathcal{S}_5}$. This was shown by Donagi in 1987, and in a second paper that year, he offered a solution:

Let ${S_g^{\mathrm{big}}}$ be what was earlier called ${\mathcal{S}_g}$, and we define ${ \mathcal{S}_g^{\mathrm{small}}}$ to be the image of the intersection of the translates of ${\mathcal{RS}_g}$ under the points of order 2, that is, ${\mathcal{S}_g^{\mathrm{small}}}$ is the locus of abelian varieties whose fiber from ${\mathcal{RS}_g}$ is everything. The new conjecture then becomes ${\mathcal{S}_g^{\mathrm{small}}=\mathcal{J}_g}$, and this is still believed to be true.

5. Genus 0-3

Genus ${g\leq 3}$ is fairly uninteresting, as ${\mathcal{J}}$ turns out to be dominant, as is shown by a simple dimension count for ${\mathcal{M}_g}$ and ${\mathcal{A}_g}$.

6. Genus 4

Our approach is different from Igusa’s. His was hands on, ours is based on the fibers of ${\mathcal{P}}$ and ${\beta}$. First, we set ${\mathcal{C}\subset\mathcal{A}_5}$ to be the locus of intermediate Jacobians of cubic threefolds, and this breaks into ${\mathcal{RC}^0}$ and ${\mathcal{RC}^1}$ in ${\mathcal{RA}_5}$, the even and odd parts (odd and even refers to the dimension of a certain cohomology group). It turns out that there is a map ${\kappa:\mathcal{A}_4\rightarrow \mathcal{RC}^0}$ which is birational.

Now, if ${A\in\mathcal{A}_4}$, the fiber over ${A}$ of ${{\mathcal{P}}$ is ${\widetilde{F(\kappa(A))}}$, a double cover of the Fano surface of lines in ${\kappa(A)}$. This turns out to let us prove that ${\beta^{-1}(J(C))}$ is, away from the boundary (which we ignore in this case), just ${K(J(C))}$ for ${C\in\mathcal{M}_3}$, and so is irreducible. As ${\mathrm{im}\alpha}$ is as well, we know that ${\beta^{-1}(\mathrm{im}\alpha)}$ is.

7. Genus 5

Genus 5 is current work inprogress. First we look at ${\mathcal{P}:\mathcal{RM}_6\rightarrow\mathcal{A}_5}$. It turns out that ${\deg\mathcal{P}=27}$. As it happens, the tetragonal construction furnishes an incidence relation on the fibers, and it is equivalent to that of lines on a cubic surface.

Using this, we can show that ${\beta}$ is also finite, and of degree 119.

Claim (Almost a theorem): ${\mathcal{RS}_5=\mathcal{RJ}_5\cup (\mathcal{A}_4\times\mathcal{RA}_1)\cup\mathcal{RC}^0\cup \partial^I\mathcal{RA}_5}$ in the toroidal partial compactification. (Here, ${\partial^I}$ denotes the component of the boundary where the vanishing cycle is ${\mod 2}$ orthogonal to the point of order 2)

Partial Proof: We can compute the degree of ${\beta}$ on each component. ${\deg\beta|_{\mathcal{RJ}_5}=54=2*27}$, ${\deg\beta|_{\mathcal{RC}^0}=1}$, ${\deg \beta|_{\mathcal{A}_4\times\mathcal{RA}_1}=0}$ and ${\deg\beta|_{\partial^I\mathcal{RA}_5}=64}$, so all that remains is a computation to show that nothing else appears which contributes 0.

This claim then implies ${\mathcal{S}_5^{\mathrm{small}}=\mathcal{J}_5}$. 