# Scattone's Baily Borel compactifications

## Degree $2d$ compactifications

:::{.remark}
The main reference for this section is \cite{scattone1987on-the-compactification-of-moduli}, which describes the Baily-Borel compactifications of $F_{2d}$. 
The main result of this work is to describe $\partial\bbcpt{F_{2d}}$ using lattice-theoretic techniques, giving partial cusp diagrams for certain arithmetically constrained values of $d$.
In particular, it shows that the number of 1-cusps is asymptotic to $d^8$, and the number of 0-cusps is 1 when $d$ is squarefree, and otherwise is given by the function ???.
Complete details are given for the cases $d=1,2$.

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:::{.remark}
One first notes that by the global Torelli theorem for algebraic K3 surfaces \cite{PSS71}, $F_{2d}$ admits a coarse space of the form $D_{L_d}/\Gamma_{L_d}$ for a certain choice of lattice $L_d$.
Note that $D_{L_d}$ is a 19-dimensional bounded symmetric domain of type IV and $\Gamma_{L_d}$ is an arithmetic group acting upon it.
Recall that the theory of automorphic forms realizes $\bbcpt{D_{L_d}/\Gamma_{L_d}}$ as a projective variety.
If $L_d$ is the primitive cohomology of a polarized K3 surface, there is a correspondence between $n$-dimensional boundary cusps and $\Gamma_{L_d}$-orbits of $n+1$-dimensional isotropic subspaces in $L_d$.
The boundary of the Baily-Borel compactification is "small" in the sense that it has very high codimension, and thus the geometric information it contains is insufficient to reconstruct a birationally unique family from a family over the punctured disc.


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:::{.remark}
A polarization of degree $2d$ on a K3 surface is a primitive divisor $H$ with $H^2 = 2d > 0$ which is pseudoample, i.e. $HD\geq 0$ for any effective $D\in \Div(X)$.
A primitively polarized K3 surface is a pair $(X, H)$.
Choose a marking $\phi: H^2(X; \ZZ)\to \lkt$ such that $\phi(H) = h$ where $h$ is a fixed primitive vector satisfying $h^2=2d$.
So $\Orth(\lkt)$ acts transitively on the set of primitive vectors of a fixed square, such an isometry can always be found and the choice of $h$ is irrelevant.
Note that $(H, \omega_X) = 0$, and thus the period of $(X, H)$ lies in $\Omega_{2d} \da \Omega_S$ for $S \da h^{\perp \lkt}$.
Set $\Gamma_{2d} \da \Stab_{\Orth(\lkt)}(h)$; then $\Gamma_{2d}\actson \Omega_{2d}$ discontinuously and $\Omega_{2d}/\Gamma_{2d}$ is a normal complex analytic space which serves as a coarse space for $F_{2d}$ by \cite{PSS71,friedman1984a-new-proof}.
Because any two choices of $h$ are equivalent modulo $\Orth(\lkt)$, the isomorphism class of $h^{\perp \lkt}$ depends only on $d$. 
Making an appropriate choice of $h$, one can identify
\[
L_{2d} \da h^{\perp \lkt} \cong \gens{-2d} \oplus U^{\oplus 2} \oplus E_8^{\oplus 2}
.\]
One can now define the period domain as $\Omega_{2d} \da \Omega_{L_{2d}}$.
This consists of two connected components interchanged by conjugation, so $\Omega_{2d} = D_{2d} \union \tilde D_{2d}$, and we fix once and for all a choice of one component which we will denote $D_{2d}$.
It is well known that
\[
D_{2d} \cong {\SO^0_{2, 19} \over \SO_2 \times \SO_{19} }
.\]
We set $\tilde \Orth(L_{2d})$ to be the image of $\Gamma_{2d}$ under the injection $\Gamma_{2d}\injects \Orth(L_{2d})$ induced by restriction -- note that this coincides with the general definition $\tilde \Orth(L) \da \ker(\Orth(L) \to \Orth(q_L))$.
We set $\Orth_-(L_{2d})$ to be the index 2 subgroup that preserves the component $D_{2d}$, and $\Gamma_{2d} \da \tilde \Orth(L_{2d}) \intersect \Orth_-(L_{2d})$, we obtain identifications
\[
F_{2d} \cong \Omega_{2d}/\tilde\Orth(L_{2d}) = D_{2d}/\Gamma_{2d}
.\]
We focus our attention on the latter definition, $F_{2d} \da D_{2d}/\Gamma_{2d}$, and more generally on compactifications of general $D/\Gamma$.


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## General theory

:::{.remark}
Let $D$ be a symmetric bounded domain and $\Gamma \leq \Aut(D)$ a discrete arithmetic subgroup of automorphisms.
Equivalently, we can write $D = G(\RR)/K$ for $G$ a connected linear algebraic group defined over $\QQ$ and $K$ a maximal compact subgroup of $G(\RR)$.
We then require that $\Gamma \leq G$ is arithmetic, i.e. $\Gamma \subseteq G(\QQ)$ and is commensurable with $G(\ZZ)$.
We will generally define $\overline{D/\Gamma}$ as $D^*/\Gamma$ where $D \subseteq D^* \subseteq D\dual$ is a subset of the compact dual via the Borel embedding, comprised of $D$ and rational boundary components.


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:::{.remark}
Regarding $D \subseteq D\dual$, we have $\partial D = \disjoint F_i$ where each $F_i$ is a boundary component, i.e. a maximal connected complex analytic set.
We set 
\[
N_F \da \ts{g\in G(\RR) \st gF = F} \da \Stab_{G(\RR)}(F)
\]
to be the stabilizer of a boundary component and note that the maximal parabolic subgroups of $G(\RR)$ are precisely those of the form $N_F$.
A boundary component is rational when $N_F(\CC)$ is defined over $\QQ$.
Let $B(D)$ be the set of proper rational boundary components of $D$.
Then there is a bijection
\begin{align*}
B(D) &\mapstofrom \ts{\text{Proper maximal parabolic $\QQ$-subgroups of } G(\CC)} \\
F &\mapstofrom N_F(\CC)
\end{align*}

We can write 
\[
D^* = D\union \Disjoint_{F\in B(D)} F
.\]
Then $\overline{D/\Gamma} \da D^*/\Gamma$ can be written as 
\[
D/\Gamma \union \coprod_{[F]\in B(D)/\Gamma} V_F
\]
where $V_F$ are varieties and we index over orbits of rational boundary components modulo $\Gamma$.

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:::{.remark}
We can identify the $V_F$ explicitly: write $G_F \da \Stab_{G(\RR)}(F)/\Fix_{G(\RR)}(F)$ and $N_{\Gamma, F} \da \Stab_\Gamma(F)/\Fix_\Gamma(F)$, then $V_F = F/N_\Gamma(F)$.
Note that in applications to K3 surfaces, we have $G(\RR) = \SO^0_{2, 19}$.
In this situation, we have a correspondence
\begin{align*}
\partial D_L &\mapstofrom \OGr(L_\RR) \\
F &\mapstofrom E
\end{align*}
where $E$ corresponds to $F$ iff $\Stab_{\Orth(L_\RR)}(E) = \Stab_{G(\RR)}(F) \da N_F$.
Restricting to rational boundary components corresponds to $\OGr(L_\QQ)$, which are further identified with $\OGr(L)$, the primitive isotropic sublattices of $L$.
For any subgroup $\Gamma_L \leq G(\ZZ)$ we obtain a bijection
\[
B(D_L)/\Gamma_L \mapstofrom \OGr(L)/\Gamma_L
\]
which preserves incidence relations.

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:::{.example}
As an example, one can take the symplectic form on $\ZZ^{2g}$, which yields $G(\RR) = \PP\Sp_g(\RR)$ and $D$ is the Siegel upper half space $\cH^g$.
Let $\Gamma = G(\ZZ) = \PP \Sp_g(\ZZ)$ be the full Siegel modular group.
Then maximal parabolic $\QQ$-subgroups of $G_\RR$ correspond to stabilizers of rational isotropic subspaces of $L_\RR$.
Let $H^n \da \cH^n/\Sp_n(\ZZ)$, then $\overline{D/\Gamma} = H_g \union H_{g-1}\union \cdots \union H_1\union H_0$ coincides with the Satake compactification.

Consider now the example of $F_{2d}$.
Recall that $D$ is a component of $\Omega \da \ts{z\in \PP L_\CC \st z^2 = 0,\, z\bar z > 0}$ and $D\dual = \ts{z\in \PP L_\CC \st z^2 = 0}$ is a quadric.
Stratify $\partial D = \partial_1 D \disjoint \partial_0 D$, noting that $\partial_0 D = D\dual \intersect \PP L_\RR$.
Then all points of $\partial_0 D$ are of the form $\PP(\gens{v}_\CC)$ where $v\in L_\RR, v^2 = 0$ is isotropic.
All components in $\partial_1 D$ are of the form $\PP(\gens{v, w}_\CC) \intersect \partial_1 D$ where $\gens{v, w}_\RR$ varies in $\OGr_2(L_\RR)$.
Restricting to rational components, one considers $\OGr_2(L)$ instead, i.e. subspaces $E_\RR$ arising from sublattices $E \leq L$.
We thus obtain bijections
\begin{align*}
B_0(D) &\mapstofrom \OGr_1(L) \\
\PP\gens{v}_\CC &\mapstofrom \gens{v}_\ZZ
\end{align*}
and
\begin{align*}
B_1(D) &\mapstofrom \OGr_2(L) \\
\PP\gens{v, w}_\CC \intersect \partial_1 D &\mapstofrom \gens{v, w}_\ZZ
\end{align*}

One can then write
\begin{align*}
\overline{D/\Gamma} = 
D/\Gamma \union
\coprod_{[\gens{v}_\ZZ] \in \OGr_1(L)/\Gamma } p_v \union
\coprod_{[\gens{v, w}_\ZZ] \in \OGr_2(L)/\Gamma } {\PP\gens{v, w}_\CC \intersect \partial_1 D \over N_\Gamma(\gens{v, w}_\ZZ)}
\end{align*}
where $N_\Gamma(E) = \Stab_\Gamma(E) / \Fix_\Gamma(E)$ can be identified with the image of $\Stab_\Gamma(E)$ in $\SL(E) \cong \SL_2(\ZZ)$.

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