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)[2]\}=} {\{(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[2]\}}

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}.

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About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
This entry was posted in Abelian Varieties, Conferences, Curves, Hodge Theory, ICTP Summer School, Moduli of Curves. Bookmark the permalink.

One Response to The Schottky Problem (ICTP)

  1. Pingback: Prym Varieties « Rigorous Trivialities

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