# The Baily-Borel compactification

:::{.remark}
The Baily-Borel and toroidal compactifications are defined for quotients of Hermitian symmetric spaces by actions of arithmetic subgroups of their automorphism groups, i.e. those that can be written as $\dcosetl{\Gamma}{\Omega}$. BB compactifications are generally small, e.g. $\dim F_2 = 19$ but $\codim \partial \bbcpt{F_2} = 18$, and this often precludes having a satisfactory modular interpretation of its boundary points. In particular, given an arc in this compactification with endpoint in the boundary, one can not generally construct a birationally unique limit. 
Toroidal compactifications $\torcpt{\dcosetl{\Gamma}{\Omega}}$ are obtained as certain blowups of $\bbcpt{\dcosetl{\Gamma}{\Omega}}$, and e.g. for $F_2$ some boundary components become divisors (codimension 1). However these are highly non-unique and depend on choices of fans. One might hope there are canonical such choices. The semitoric compactifications of Looijenga interpolate between $\bbcpt{\dcosetl{\Gamma}{\Omega}}$ and $\torcpt{\dcosetl{\Gamma}{\Omega}}$.
:::


:::{.definition}
The group $G$ acts transitively on the set of boundary components $F \subseteq \partial \cD_L \da \tilde\cD_L\setminus \cD_L$, and $\Stab_G(F)\leq G$ is a maximal parabolic subgroup. 
Taking stabilizers establishes a bijection
\begin{align*}
\ts{\text{Boundary components $F\subseteq \partial \cD_L$}} &\to \ts{\text{Maximal parabolic subgroups $P\leq G$}} \\
F &\mapsto P_F \da \Stab_G(F)
\end{align*}

For $G \da \SO(V)$, parabolic subgroups $P$ are stabilizers of flags of isotropic subspaces in $V$, and since $\signature(V) = (2, n)$, a flag has length at most 3 and a maximal flag is of the form $p \subseteq I\subseteq J$ where $p$ is a point, $I$ is an isotropic line, and $J$ is an isotropic plane. The only flags that define maximal parabolic subgroups of $\SO(V)$ are of length 1, consisting of either a single line or a single plane. Thus we have bijections
\begin{align*}
\ts{\text{Rational boundary components of }
(\SO_{2, n}(\RR), \SO_2(\RR) \times \SO_n(\RR))
} \\
\reflectbox{\rotatebox[origin=c]{90}{$\mapstofrom$}}\hspace{10em} \\
\ts{\text{Maximal parabolic subgroups of }
\SO(V)
} \\
\reflectbox{\rotatebox[origin=c]{90}{$\mapstofrom$}}\hspace{10em} \\
\ts{\text{Isotropic lines $I\subset V$ }}
\union
\ts{\text{Isotropic planes $J \subset V$ }}
\end{align*}
where a boundary component $F$ is rational if $\Stab_G(F)$ is defined over $\QQ$.
For an arithmetic subgroup $\Gamma \leq G$, letting $\partial(\cD_L)_\QQ$ be the set of all rational boundary components of $\cD_L \subset \tilde \cD_L$, we produce a compactification
\[
\overline{ \dcosetl{\Gamma}{\cD_L} }
= 
\dcosetl{\Gamma}{\cD_L} 
\Union_{F\in \partial(\cD_{L})_\QQ } 
\dcosetl{(G_F(\QQ) \intersect \Gamma)}{F}
\]

:::

:::{.definition}
Let $L$ be a lattice of signature $(2, n)$ for $n\geq 1$, let $\Omega_L$ be the associated period domain, let $\Orth^+(L) \leq \Orth(L)$ be the subgroup preserving $\Omega_L$ and let $\tilde \Omega_L$ be the affine cone over $\Omega_L$.
Let $n\geq 3$, let $k\in \ZZ$, and let $\Gamma \leq \Orth^+(L)$ be a finite-index subgroup with $\chi: \Gamma\to \CC^*$ a character. A holomorphic functional $f: \Omega_L \to \CC$ is called a \textbf{modular form of weight $k$ and character $\chi$ for $\Gamma$} if

- Factor of automorphy: $f(\lambda z) = \lambda^{-k} f(z)$ for any $\lambda \in \CC^*$.
- Equivariance: $f(\gamma z) = \chi(\gamma) f(z)$ for all $\gamma\in \Gamma$.

:::

:::{.definition}
Let $\Omega_L$ as above and let $M_k(\Gamma, \chi)$ be the $\CC\dash$vector space of such modular forms of weight $k$ for $\Gamma$ with character $\chi$. The \textbf{Baily-Borel compactification} can be defined as 
\[
\bbcpt{\dcosetl{\Gamma}{\Omega_L}}
\da 
\Proj \bigoplus_{k\geq 1} M_k(\Gamma, \chi_{\mathrm{triv}})
\]
where $\chi_{\mathrm{triv}}$ is the trivial character.
:::

:::{.remark}
$\partial \bbcpt{\dcosetl{\Gamma}{\Omega_L}}$ decomposes into points $p_i$ and curves $C_j$, which are in bijection with $\Gamma\dash$orbits of isotropic lines $i$ and isotropic planes $j$ in $L_\QQ$. Moreover $p_i \in \overline{C_j} \iff$ one can choose representatives lines $i$ and planes $j$ such that $i \subseteq j$.
:::

:::{.remark}
A theorem of Baily-Borel gives the existence of an ample automorphic line bundle $\cL$ on $\overline{\cD_L}$ giving it the structure of a normal projective variety isomorphic to a canonical model $\Proj \bigoplus_{k\geq 0} H^0(L^k)??$.\dzg{I don't quite remember what this graded ring is.}
We denote this compactification $\bbcpt{\dcosetl{\Gamma}{\cD_L}}$.
:::

## Toroidal and semitoroidal compactifications

:::{.definition}
A \textbf{toroidal compactification} $\torcpt{\dcosetl{\Gamma }{\cD_L}}$ is a certain blowup of $\bbcpt{\dcosetl{\Gamma}{\cD_L}}$, so there is a birational map $\torcpt{\dcosetl{\Gamma}{\cD_L}} \rational \bbcpt{\dcosetl{\Gamma}{\cD_L}}$. It is defined by a collection of admissible fans $\ts{F_i}_{i\in I}$ where $I$ ranges over an index set for all cusps.
:::

:::{.definition}
A \textbf{semitoroidal compactification} is a generalization due to Looijenga for which the cones of $F_i$ are not required to be finitely generated.
:::

:::{.remark}
\cite{AE21} shows that semitoroidal compactifications are characterized as exactly the normal compactifications $\semitorcpt{M}$ fitting into a tower 

% https://q.uiver.app/?q=WzAsMyxbMCw0LCJcXGJiY3B0e019ICJdLFswLDIsIlxcc2VtaXRvcmNwdHtNfSAiXSxbMCwwLCJcXHRvcmNwdHtNfSJdLFswLDEsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEsMiwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=
\[\begin{tikzcd}
{\torcpt{M}} \\
\\
{\semitorcpt{M} } \\
\\
{\bbcpt{M} }
\arrow[hook, from=5-1, to=3-1]
\arrow[hook, from=3-1, to=1-1]
\end{tikzcd}\]

where $\torcpt{M}$ is some toroidal compactification of $M$.

> Promising stuff here: https://dept.math.lsa.umich.edu/~idolga/EnriquesOne.pdf#page=567&zoom=160,-136,765

:::

:::{.remark}
On the toroidal compactification associated with $\dcosetl{\Gamma}{\Omega_L}$: for a cusp $C_i$ of the BB compactification, let $F\subset L_\QQ$ be the corresponding $\Gamma\dash$orbit of an isotropic line or plane. 
Consider its stabilizer $S(F) \da \Stab_{\Orth^+(L_\RR)}(F)$, and its unipotent radical $U(F)$. Then $U(F)$ is a vector space containing a lattice $U(F) \intersect \Gamma$ and an open convex cone $C(F)$. 
Let $\rc{C(F)}$ be the rational closure of the cone, so the union of $C(F)$ and rational rays in its closure. 
We then choose a fan $\Sigma(F)$ with $\supp(\Sigma(F)) = \rc{C(F)}$ which is invariant under $S(F) \intersect \Gamma$ and produce an associated toric variety $X_{\Sigma(F)}$. If one does this for every $F$ to produce a $\Gamma\dash$admissible collection of polyhedra $\Sigma$, their quotients by $\Gamma$ glue to give a toroidal compactification $\torcpt{\dcosetl{\Gamma}{\Omega_L}}$, which has the structure of a (complex) algebraic space.
There is a surjection $\torcpt{\dcosetl{\Gamma}{\Omega_L}} \surjects \bbcpt{\dcosetl{\Gamma}{\Omega_L}}$.
Why $e^\perp/e$ shows up: if $e$ is an isotropic line in $L$ corresponding to a cusp $F$, there is an isomorphism of lattices $U(F) \intersect \tilde \Orth^+(L) \cong e^\perp/e$ where $\tilde \Orth^+ \da \ker\qty{\Orth^+(L) \to \Orth(\discgroup L)}$.
:::


## Misc

:::{.definition title="Log CY pairs"}
A \textbf{log Calabi-Yau (CY) pair} is a pair $(X, D)$ with $X$ a proper variety and $D$ an effective $\QQ\dash$Cartier divisor such that the pair is log canonical and $K_X + D \sim_\QQ 0$.
:::

:::{.definition}
A degeneration $\pi: \cX \to \Delta$ is a **CY degeneration** if $\pi$ is proper, $K_{\cX}\sim_\QQ 0$, and $(\cX, \cX_0)$ is dlt. This implies that $\cX_t$ is a Calabi-Yau variety for $t\neq 0$ and $\cX_0$ is a union of log CY pairs $(V_i, D_i)$. If $\cX_t$ is a strict CY of dimension $n$, so $\pi_1 \cX_t = 0$ and $h^i(\cX_t, \OO_{\cX_t}) = 0$ for $1 \leq i \leq n-1$, and $\dim \Gamma(\cX_0) = n$, we say $\cX$ is a large complex structure limit or equivalently a maximal unipotent or MUM degeneration. 
:::

:::{.remark}
If $n = 2$, Kulikov shows that $\Gamma(\cX_0)$ is always isomorphic to a 2-sphere $S^2$. Whether $\Gamma(\cX_0) \cong S^n$ or a quotient thereof for $n\geq 3$ more generally is an open question, posed by Kontsevich-Soibelman. It has recently been shown by Kollár-Xu that in the case of degenerations of Calabi-Yau or hyperkähler manifolds, the dual complex is always a rational homology sphere.
:::

:::{.remark}
Why this is useful to us: one formulation of mirror symmetry is the formulation due to Strominger-Yau-Zaslow, aptly called SYZ mirror symmetry. Conjecturally, the general fiber $\cX_t$ of a punctured family of CYs $\cX^\circ \to \Delta^\circ$ can be given the structure of a special Lagrangian torus fibration $\cX_t \to B$, one can "dualize" the fibration to obtain a mirror CY $\hat{\cX_t} \to B$ over the same base.\dzg{This might have something to do with the discrete Legendre transform Phil mentions.} The common base $B$ of these two fibrations is conjecturally of the form $\Gamma(\cX_0)$, the dual complex of a degeneration $\cX\to \Delta$ extending $\cX^\circ$.
:::


## Baily-Borel cusps and incidence diagrams


## Other compactifications

:::{.remark}

> See https://arxiv.org/pdf/2010.06922.pdf#page=1&zoom=auto,-17,32

Write $F_{2d} \da \dcosetl{\Gamma_{2d}}{D_{2d}}$.
A cusp $p_i$ of $\bbcpt{F_{2d}}$ determines a cone $C_i$.
Toroidal and semitoroidal compactifications are then determined by a collection of $\Gamma_{2d}\dash$invariant fans supported on $C_i$ for $i$ ranging over an index set for all cusps.
If $d=1$ (or more generally if $2d$ is squarefree), there is a single 0-cusp $p_{2d}$ whose cone $C_{2d}$ has a description as the positive light cone in the rational closure of $C_{2d}$ with respect to a certain lattice $M_{2d}$. This can be written $C^{\mathrm rc }_{2d} \da \operatorname{Conv}(\overline{C}_{2d} \intersect M_{2d}\tensor_\ZZ \RR)$. 
A semitoroidal compactification of $F_{2d}$ is then determined by a semitoric fan in $M_{2d, \RR}$ supported on $C_{2d}^{\mathrm rc}$ which is invariant for a particular subgroup $\Gamma^+_{2d} \leq \Orth(M_{2d})$.
\dzg{Todo: can spell out what $M_{2d}, \Gamma, \Gamma^+$ are.} In this case, one can make a canonical choice for such a semitoric fan: the Coxeter fan $\Sigma_{2d}^{\mathrm{Cox} }$ whose cones are precisely the fundamental domains for a Weyl group action on $C_{2d}^{\mathrm rc}$, see AET19.
:::