---
created: 2024-05-03T00:13
updated: 2024-05-04T21:36
---

# 8-24 

Let us quickly review our setup: consider $U(2) \subseteq U(2) + E_8(2)$, and take its perp to obtain $L_- \subseteq \Lambda_{18} \da U\sumpower{2} + D_{16}$. Let's call the associated period domain $\Omega_2$, we then want a morphism $\Omega_2 \embeds \Omega_{4, h.e.}$, where we have group actions $\Lambda_2 \actson \Omega_2$ and $\Lambda_{4, h.e.}\da \Orth(\Lambda_{18})\actson \Omega_{4, h.e.}$. Note that $\Gamma_2 \neq \Orth(L_-)$!

Is there an induced morphism on the quotients $\Omega_2/\Gamma_2 \to \Omega_{4, h.e.} /\Gamma_{4, h.e.}$? The answer is yes, and injectivity follows from applying the Torelli theorem and using a geometric argument. As a result, we get an embedding of quasiprojective varieties $\phi: E_2 \embeds F_{4, h.e.}$. Applying an extension result that Luca found and added to the paper, in this case we do get an extension of this morphism to the BB compactifications, 
$$
\bar\phi: \overline{ E_{2}{}^{BB}} \to \overline{ F_{4, h.e.}{}^{BB}}
$$
and Luca proceeded to study the cusp correspondence.

Question: is there a morphism $\Gamma_2 \to \Gamma_{4, h.e.}$?

A neat trick: for any lattice $L$, there is a well-defined diagonal map 
$$
\phi: L(2) \to L\oplus L \quad x \mapsto (x,x)
$$
Why this is true: 
$$(\phi(x), \phi(x))_{L\oplus L} = ((x, x), (x, x))_{L\oplus L} = (x, x)_L + (x, x)_L = 2(x, x)_L = (x, x)_{L(2)}$$

In what follows, we take $e\in L_- \subseteq \Lambda_{18}$ and consider $e^{\perp L_-}/e \subseteq e^{\perp \Lambda_{18}}/e$ and make identifications $U(2) + E_8(2) \subseteq U(2) + E_8^2$ and $U + E_8(2) \subseteq U + E_8^2$.
## Question 1a

In the Laza-O'Grady paper, where to they compute $e^\perp/e$? We believe $e^\perp/e \cong U(2) + E_8^2$ or $U + E_8^2$; we should look into theorem 2.8 for the definitions of the $\rm{III}_a$ and $\rm{III}_b$ conventions. Is this computed more explicitly in one of Valery's papers? Or Scattone? 

We do know that in Sterk's paper, he shows $e^\perp/e\cong U(2) + E_8(2)$ or $U + E_8(2)$.

## Question 1b

We have the morphism
$$
\bar\phi: \overline{ E_{2}{}^{BB}} \to \overline{ F_{4, h.e.}{}^{BB}}
$$
But we are not so sure it is injective. What Luca can say for sure is that the restriction of this map onto its image is in fact the normalization.

## Question 2

Luca has found an abstract extension result that describes when a morphism symmetric spaces lifts to a morphism on their BB compactifications. It is somewhat abstract and not very symmetric.

I mentioned here that there is some interest from people in the UK who study a reverse problem: given a period domain cooked up from a lattice, does it "come from geometry"? They are usually classifying spaces of Hodge structures, but what spaces realize those?

## Question 3

We would like to know the actual presentations of the lattices of the form $\pi^\perp/\pi$ where $\pi$ is an isotropic plane for $E_2$; Sterk describes 9 of these.

> There is more we talked about, but I haven't recorded it here yet.