# Moduli of polarized K3 surfaces of degree $2d$ 

:::{.remark title="K3 surfaces"}
A **K3 surface** is a smooth projective surface with trivial canonical bundle $\omega_X \cong \OO_X$ and $h^1(\OO_X) = 0$. Prototypical examples include double branched covers of sextic curves in $\PP^2$ and smooth quartic hypersurfaces in $\PP^3$. 
All K3 surfaces are diffeomorphic, and thus have the Hodge diamond shown in \cref{fig:hodge-k3-surface}.

\begin{figure}[H]
\centering
\begin{tikzcd}[column sep=small,row sep=small]
	&& {h^{2,2}} &&&&&& 1 \\
	& {h^{2,1}} && {h^{1,2}} &&&& 0 && 0 \\
	{h^{2,0}} && {h^{1,1}} && {h^{0,2}} & {=} & 1 && 20 && 1 \\
	& {h^{1,0}} && {h^{0,1}} &&&& 0 && 0 \\
	&& {h^{0,0}} &&&&&& 1
\end{tikzcd}
\caption{The Hodge diamond of a K3 surface.}
\label{fig:hodge-k3-surface}
\end{figure}

:::

:::{.remark title="Lattice theory in moduli"}
The cup product endows the singular cohomology $H^2(X, \ZZ)$ with the structure of a lattice, where by a **lattice** we mean a finitely generated free $\ZZ$-module with a nondegenerate $\ZZ$-valued symmetric bilinear form. It is isometric to the K3 lattice:
\[
H^2(X; \ZZ) \cong \lkt \da U\oplus U\oplus U \oplus E_8 \oplus E_8
,\]
where $U$ is the hyperbolic lattice, the unique even unimodular lattice of rank 2 with Gram matrix $\bigl( \begin{smallmatrix}0 & 1\\ 1 & 0\end{smallmatrix}\bigr)$ and $E_8$ is the negative-definite lattice associated to the $E_8$ Dynkin diagram.

By the weak Torelli theorem for K3 surfaces \cite[Cor.\, 8.1.1.2]{BHPV04}, the moduli theory of K3 surfaces is regulated by this lattice structure.
Of particular importance is the **Néron-Severi** lattice $\NS(X) \da H^{1,1}(X) \intersect H^2(X, \ZZ)$ of integral $(1,1)$ forms, and its orthogonal complement in $H^2(X, \ZZ)$, the **transcendental lattice**. We refer to these as $S_X$ and $T_X$ respectively.
By the Lefcshetz $(1,1)$ theorem \cite[Thm.\, 4.2.13]{BHPV04}, the first Chern class $c_1: \Pic(X) \to \NS(X)$ induces an isometry.
Because the naive construction of a coarse moduli space of projective K3 surfaces yields a non-Hausdorff space, we restrict our attention to **polarized K3 surfaces of degree $2d$** -- pairs $(X, L)$ where $X$ is a K3 surface and $L$ is an ample line bundle on $X$ satisfying $L^2 = 2d > 0$.

:::

:::{.remark title="Lattice polarized K3 surfaces"}
Let $S$ be a non-degenerate lattice of signature $(1, n)$ which admits a primitive embedding into $\lkt$ (which is of signature $(3, 19)$) and $T \da S^{\perp \lkt}$ which is of signature $(2, 19-n)$. 
We then define the period domain associated to $S$ as a connected component $D_S$ of
\[
\Omega_S \da \ts{ [\sigma]\in \PP\qty{T\tensor \CC} \st \sigma^2 = 0, \sigma\bar\sigma > 0}
,\]
yielding a Hermitian symmetric domain of Type IV.
Let $\Gamma_S \da \tilde \Orth(T)$ where
\[
\tilde \Orth(T) \da \ker(\Orth(T)\to \Orth(A_T))
\]
and $A_T \da T\dual/T$ is the discriminant group of $T$.
It can be shown that
\[
F_S \da \dcosetl{\Gamma_S}{D_S}
\]
is a coarse moduli space of $S$-polarized K3 surfaces.
Taking $S\da \gens{h}$ the sublattice generated by an ample class $h$ satisfying $h^2 = 2d$ recovers $F_{2d} \da F_{\gens{h}}$, noting that $\gens{h}^{\perp \lkt} \cong \gens{-2d}\oplus U^{\oplus 2}\oplus E_8^{\oplus 2}$.
By \cite{piateski-shapiro1971torelli}, $F_{2d}$ constructed in this way is a coarse moduli space of degree $2d$ primitively polarized K3 surfaces.

To such surfaces, by \cite{BB66} there is a canonically defined quasiprojective compactification $\dcosetl{\Gamma_S}{D_S} \injects \bbcpt{ \dcosetl{\Gamma_S}{D_S} }$ whose boundary consists of 0-cusps (points) and 1-cusps (curves).
The 0-cusps are in bijection with $\Gamma_S$-orbits of isotropic lines $I \subseteq T$, and the 1-cusps are in bijection with orbits of isotropic planes $J \subseteq T$.
By \cite{AMRT75}, there exists a class of compactifications $\dcosetl{\Gamma_S}{D_S} \injects \overline{\dcosetl{\Gamma_S}{D_S}}^{\mcf}$ defined by the combinatorial data of a fan $\mcf \da \ts{\mcf_I}$ where $I$ ranges over $\Gamma$-orbits of isotropic lines in $T$, equivalently as $I$ ranges over the 0-cusps of $\partial \bbcpt{\dcosetl{\Gamma_S}{D_S}}$.
These are referred to as toroidal compactifications.
Noting that $\signature(I^{\perp T}/I) = (1, 18)$, we obtain a hyperbolic lattice and can construct a model of hyperbolic space $\HH^{18}$ as a projectivization of the positive cone 
\[
C^+ \da \ts{v\in I^\perp/I\tensor \RR \st v^2 > 0}
.\]
Letting $\Gamma_{S, I} \da \Stab_{\Gamma_S}(I)$, the data of $\mcf_I$ is specified by a $\Gamma_{S, I}$-invariant rational polyhedral tiling of $\HH^{18}$.
The combinatorics of such a tiling determines a union of toric varieties which are adjoined as the boundary strata of $\dcosetl{\Gamma_S}{D_S}$ at the 0-cusp corresponding to $I$.
This produces a divisorial boundary with mild singularities.
The work of \cite{Loo86} introduces semitoroidal compactifications, allowing for the tiling to be *locally* rationally polyhedral, which simultaneously generalizes the Baily-Borel and toroidal compactifications described above.
:::

:::{.remark}
A modular alternative to these compactifications was introduced in \cite{kollar1988threefolds-and-deformations}, denoted the space of stable slc pairs, which is proper.
A pair $(X, R)$ of a projective variety $X$ with a $\QQ$-divisor $D$ is stable if $K_X + R$ is ample and $\QQ$-Cartier and $(X, R)$ has slc singularities.
These strictly generalize the stable curves that appear in $\overline{\Mgn}$, and naturally generalize these notions to all dimensions.
For an appropriately universal choice of polarizing divisor $R$ on the generic K3 surface in $F_{2d}$, one can define a compactification $\overline{F_{2d}}^{R}$ as the closure of the space of pairs $(X, \varepsilon R)$ in a Zariski open subset of $F_{2d}$ in the space of stable slc pairs.
For example, consider a degree 2 polarized K3 surface $(X, L)$. 
The linear system $\abs{L}$ induces a branched 2-to-1 cover $X\to \PP^2$, and one can choose $R$ to be the ramification divisor of the covering involution.
By ???, when $R$ is a *recognizable divisor*, there is a unique semifan $\mcf_R$ such that the normalization of $\overline{F_{2d}}^R$ is isomorphic to the semitoroidal compactification $\overline{F_{2d}}^{\mcf_R}$.
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:::{.remark title="Kulikov models"}
Any degeneration of K3 surfaces is birational to a **Kulikov model**: after a birational modification and a ramified base change, it may be put in the form of a degeneration $\pi: \cX\to \DD$ over the complex disc such that $\pi$ is semistable with trivial canonical $\omega_{\cX} \cong \OO_{\cX}$.

By ?, the central fiber of a type III Kulikov model is encoded in an integral affine 2-sphere, abbreviated $\IAS^2$.

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