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created: 2024-05-03T00:13
updated: 2024-05-11T11:38
---

::: 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
$\Gamma\backslash \Omega$. 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
$\torcpt{\Gamma\backslash \Omega}$ are obtained as certain blowups of
$\bbcpt{\Gamma\backslash \Omega}$, 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 $\bbcpt{\Gamma\backslash \Omega}$ and
$\torcpt{\Gamma\backslash \Omega}$.
:::

## The Baily-Borel compactification

::: definition
The group $G$ acts transitively on the set of boundary components
$F \subseteq \partial \DD_L \da \tilde\DD_L\setminus \DD_L$, and
$\Stab_G(F)\leq G$ is a maximal parabolic subgroup. Taking stabilizers
establishes a bijection $$\begin{aligned}
    \ts{\text{Boundary components $F\subseteq \partial \DD_L$}} &\to \ts{\text{Maximal parabolic subgroups $P\leq G$}} \\
    F &\mapsto P\da \Stab_G(F)
\end{aligned}$$ 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 $$\begin{aligned}
\ts{\text{Rational boundary components of }
(\SO_{2, n}(\RR), \SO_2(\RR) \times \SO_n(\RR))
} \\
\reflectbox{\rotatebox[origin=c]{90}{$\mapstofrom$}}\hspace{10em} \\
\ts{\text{Maximal parabolic subgroups of}
\SO(V)
} \\
\reflectbox{\rotatebox[origin=c]{90}{$\mapstofrom$}}\hspace{10em} \\
\ts{\text{Isotropic lines $I\subset V$ }}
\union
\ts{\text{Isotropic planes $J \subset V$ }}
\end{aligned}$$ where a boundary component $F$ is rational if
$\Stab_G(F)$ is defined over $\QQ$. For an arithmetic subgroup
$\Gamma \leq G$, letting $\partial(\DD_L)_\QQ$ be the set of all
rational boundary components of $\DD_L \subset \tilde \DD_L$, we produce
a compactification
$${\overline {\Gamma\backslash \DD_L}} = {\Gamma \backslash \DD_L} \Union_{F\in \partial\DD{L, \QQ} } {(G_F(\QQ) \intersect \Gamma) \backslash F}.$$
:::

::: 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 **modular form of weight $k$ and
character $\chi$ for $\Gamma$** if

1.  Factor of automorphy: $f(\lambda z) = \lambda^{-k} f(z)$ for any
    $\lambda \in \CC^*$.

2.  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 **Baily-Borel compactification** can be
defined as
$$\bbcpt{\Gamma\backslash \Omega_L} \da \Proj \bigoplus_{k\geq 1} M_k(\Gamma, \chi_{\mathrm{triv}})$$
where $\chi_{\mathrm{triv}}$ is the trivial character.
:::

::: remark
$\partial \bbcpt{\Gamma\backslash \Omega_L}$ 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 $\mcl$ on $\overline{\DD_L}$ giving it the structure of a
normal projective variety isomorphic to a canonical model
$\Proj \bigoplus_{k\geq 0} H^0(L^k)??$. We denote this compactification
$\bbcpt{\Gamma\backslash \DD_L}$.
:::

## Toroidal and semitoroidal compactifications

::: remark
A **toroidal compactification** $\torcpt{\Gamma \backslash \DD_L}$ is a
certain blowup of $\bbcpt{\Gamma \backslash \DD_L}$, so there is a
birational map
$\torcpt{\Gamma \backslash \DD_L} \rational \bbcpt{\Gamma \backslash \DD_L}$.
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.

A **semitoroidal compactification** is a generalization due to Looijenga
for which the cones of $F_i$ are not required to be finitely generated,
and [@AE21] shows that semitoroidal compactifications are characterized
as exactly the normal compactifications $\semitorcpt{M}$ fitting into a
tower

\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 $M$.
:::

::: remark
On the toroidal compactification associated with
$\Gamma\backslash \Omega_L$: 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 $S(F) \da \Stab_{\Orth^+(L_\RR)}$, 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
$\Stab_{\Orth^+(L_\RR)}(F) \intersect \Gamma$ 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
$\torcpt{\Gamma\backslash \Omega_L}$, which has the structure of a
(complex) algebraic space. There is a surjection
$\torcpt{\Gamma\backslash \Omega_L} \surjects \bbcpt{\Gamma\backslash \Omega_L}$.
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
A **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 $K_X + D \sim_\QQ 0$.

A degeneration $\pi: \mcx \to \Delta$ is a CY degeneration if $\pi$ is
proper, $K_{\mcx}\sim_\QQ 0$, and $(\mcx, \mcx_0)$ is dlt. This implies
that $\mcx_t$ is a Calabi-Yau variety for $t\neq 0$ and $\mcx_0$ is a
union of log CY pairs $(V_i, D_i)$. If $\mcx_t$ is a strict CY of
dimension $n$, so $\pi_1 \mcx_t = 0$ and $h^i(\mcx_t, \OO_{\mcx_t}) = 0$
for $1 \leq i \leq n-1$, and $\dim \Gamma(\mcx_0) = n$, we say $\mcx$ is a
large complex structure limit or equivalently a maximal unipotent or MUM
degeneration.

If $n = 2$, Kulikov shows that $\Gamma(\mcx_0)$ is always isomorphic to a
2-sphere $S^2$. Whether $\Gamma(\mcx_0) \cong S^n$ 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 $\mcx_t$ of a punctured family
of CYs $\mcx^\circ \to \Delta^\circ$ can be given the structure of a
special Lagrangian torus fibration $\mcx_t \to B$, one can \"dualize\"
the fibration to obtain a mirror CY $\hat{\mcx_t} \to B$ over the same
base. The common base $B$ of these two fibrations is conjecturally of
the form $\Gamma(\mcx_0)$, the dual complex of a degeneration
$\mcx\to \Delta$ extending $\mcx^\circ$.
:::

## Baily-Borel cusps and incidence diagrams

## Other compactifications

::: remark
Write $F_{2d} \da \Gamma_{2d}\backslash D_{2d}$. 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
$C^{\mathrm rc }_{2d} \da \mathrm{Conv}(\overline{C}_{2d} \intersect M_{2d}\tensor_\ZZ \RR)$.
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
$\Gamma^+_{2d} \leq \Orth(M_{2d})$. 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.
:::