Note: Author note
Abstract
Rough draft of my dissertationThe main reference for this section is , which describes the Baily-Borel compactifications of \(F_{2d}\). The main result of this work is to describe \(\partial\overline{ F_{2d} }^{\operatorname{BB}}\) 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 \(d=1,2\).
One first notes that by the global Torelli theorem for algebraic K3 surfaces , \(F_{2d}\) 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 \(\overline{ D_{L_d}/\Gamma_{L_d} }^{\operatorname{BB}}\) 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.
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 \(D\in \operatorname{Div}(X)\). A primitively polarized K3 surface is a pair \((X, H)\). Choose a marking \(\phi: H^2(X; {\mathbf{Z}})\to \Lambda_{\mathrm{K3}}\) such that \(\phi(H) = h\) where \(h\) is a fixed primitive vector satisfying \(h^2=2d\). So \({\operatorname{O}}(\Lambda_{\mathrm{K3}})\) 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 \(\Omega_{2d} \mathrel{\vcenter{:}}=\Omega_S\) for \(S \mathrel{\vcenter{:}}= h^{\perp \Lambda_{\mathrm{K3}}}\). Set \(\Gamma_{2d} \mathrel{\vcenter{:}}={\operatorname{Stab}}_{{\operatorname{O}}(\Lambda_{\mathrm{K3}})}(h)\); then \(\Gamma_{2d}\curvearrowright\Omega_{2d}\) discontinuously and \(\Omega_{2d}/\Gamma_{2d}\) is a normal complex analytic space which serves as a coarse space for \(F_{2d}\) by . Because any two choices of \(h\) are equivalent modulo \({\operatorname{O}}(\Lambda_{\mathrm{K3}})\), the isomorphism class of \(h^{\perp \Lambda_{\mathrm{K3}}}\) depends only on \(d\). Making an appropriate choice of \(h\), one can identify \begin{align*} L_{2d} \mathrel{\vcenter{:}}= h^{\perp \Lambda_{\mathrm{K3}}} \cong \left\langle{-2d}\right\rangle \oplus U^{\oplus 2} \oplus E_8^{\oplus 2} .\end{align*} One can now define the period domain as \(\Omega_{2d} \mathrel{\vcenter{:}}=\Omega_{L_{2d}}\). This consists of two connected components interchanged by conjugation, so \(\Omega_{2d} = D_{2d} \cup\tilde D_{2d}\), and we fix once and for all a choice of one component which we will denote \(D_{2d}\). It is well known that \begin{align*} D_{2d} \cong {{\operatorname{SO}}^0_{2, 19} \over {\operatorname{SO}}(2) \times {\operatorname{SO}}_{19} } .\end{align*} We set \(\tilde {\operatorname{O}}(L_{2d})\) to be the image of \(\Gamma_{2d}\) under the injection \(\Gamma_{2d}\hookrightarrow{\operatorname{O}}(L_{2d})\) induced by restriction – note that this coincides with the general definition \(\tilde {\operatorname{O}}(L) \mathrel{\vcenter{:}}=\ker({\operatorname{O}}(L) \to {\operatorname{O}}(q_L))\). We set \({\operatorname{O}}_-(L_{2d})\) to be the index 2 subgroup that preserve the component \(D_{2d}\), and \(\Gamma_{2d} \mathrel{\vcenter{:}}=\tilde {\operatorname{O}}(L_{2d}) \cap{\operatorname{O}}_-(L_{2d})\), we obtain identifications \begin{align*} F_{2d} \cong \Omega_{2d}/\tilde{\operatorname{O}}(L_{2d}) = D_{2d}/\Gamma_{2d} .\end{align*} We focus our attention on the latter definition, \(F_{2d} \mathrel{\vcenter{:}}= D_{2d}/\Gamma_{2d}\), and more generally on compactifications of general \(D/\Gamma\).
Let \(D\) be a symmetric bounded domain and \(\Gamma \leq \mathop{\mathrm{Aut}}(D)\) a discrete arithmetic subgroup of automorphisms. Equivalently, we can write \(D = G({\mathbf{R}})/K\) for \(G\) a connected linear algebraic group defined over \({\mathbf{Q}}\) and \(K\) a maximal compact subgroup of \(G({\mathbf{R}})\). We then require that \(\Gamma \leq G\) is arithmetic, i.e. \(\Gamma \subseteq G({\mathbf{Q}})\) and is commensurable with \(G({\mathbf{Z}})\). We will generally define \(\overline{D/\Gamma}\) as \(D^*/\Gamma\) where \(D \subseteq D^* \subseteq D {}^{ \vee }\) is a subset of the compact dual via the Borel embedding, comprised of \(D\) and rational boundary components.
Regarding \(D \subseteq D {}^{ \vee }\), we have \(\partial D = {\textstyle\coprod}F_i\) where each \(F_i\) is a boundary component, i.e. a maximal connected complex analytic set. We set \begin{align*} N_F \mathrel{\vcenter{:}}=\left\{{g\in G({\mathbf{R}}) {~\mathrel{\Big\vert}~}gF = F}\right\} \mathrel{\vcenter{:}}={\operatorname{Stab}}_{G({\mathbf{R}})}(F) \end{align*} to be the stabilizer of a boundary component and note that the maximal parabolic subgroups of \(G({\mathbf{R}})\) are precisely those of the form \(N_F\). A boundary component is rational when \(N_F({\mathbf{C}})\) is defined over \({\mathbf{Q}}\). Let \(B(D)\) be the set of proper rational boundary components of \(D\). Then there is a bijection \[\begin{align*} B(D) &\rightleftharpoons\left\{{\text{Proper maximal parabolic ${\mathbf{Q}}$-subgroups of } G({\mathbf{C}})}\right\} \\ F &\rightleftharpoons N_F({\mathbf{C}}) \end{align*}\]
We can write \begin{align*} D^* = D\cup\amalg_{F\in B(D)} F .\end{align*} Then \(\overline{D/\Gamma} \mathrel{\vcenter{:}}= D^*/\Gamma\) can be written as \begin{align*} D/\Gamma \cup\coprod_{[F]\in B(D)/\Gamma} V_F \end{align*} where \(V_F\) are varieties and we index over orbits of rational boundary components modulo \(\Gamma\). We can identify the \(V_F\) explicitly: write \(G_F \mathrel{\vcenter{:}}={\operatorname{Stab}}_{G({\mathbf{R}})}(F)/ \mathrm{Fix}_{G({\mathbf{R}})}(F)\) and \(N_{\Gamma, F} \mathrel{\vcenter{:}}={\operatorname{Stab}}_\Gamma(F)/ \mathrm{Fix}_\Gamma(F)\), then \(V_F = F/N_\Gamma(F)\). Note that in applications to K3 surfaces, we have \(G({\mathbf{R}}) = {\operatorname{SO}}^0_{2, 19}\). In this situation, we have a correspondence \[\begin{align*} \partial D_L &\rightleftharpoons{\operatorname{OGr}}(L_{\mathbf{R}}) \\ F &\rightleftharpoons E \end{align*}\] where \(E\) corresponds to \(F\) iff \({\operatorname{Stab}}_{{\operatorname{O}}(L_{\mathbf{R}})}(E) = {\operatorname{Stab}}_{G({\mathbf{R}})}(F) \mathrel{\vcenter{:}}= N_F\). Restricting to rational boundary components corresponds to \({\operatorname{OGr}}(L_{\mathbf{Q}})\), which are further identified with \({\operatorname{OGr}}(L)\), the primitive isotropic sublattices of \(L\). For any subgroup \(\Gamma_L \leq G({\mathbf{Z}})\) we obtain a bijection \begin{align*} B(D_L)/\Gamma_L \rightleftharpoons{\operatorname{OGr}}(L)/\Gamma_L \end{align*} which preserves incidence relations.
As an example, one can take the symplectic form on \({\mathbf{Z}}^{2g}\), which yields \(G({\mathbf{R}}) = {\mathbf{P}}{\mathsf{Sp}}_g({\mathbf{R}})\) and \(D\) is the Siegel upper half space \(\cH^g\). Let \(\Gamma = G({\mathbf{Z}}) = {\mathbf{P}}{\mathsf{Sp}}_g({\mathbf{Z}})\) be the full Siegel modular group. Then maximal parabolic \({\mathbf{Q}}\)-subgroups of \(G_{\mathbf{R}}\) correspond to stabilizers of rational isotropic subspaces of \(L_{\mathbf{R}}\). Let \(H^n \mathrel{\vcenter{:}}=\cH^n/{\mathsf{Sp}}_n({\mathbf{Z}})\), then \(\overline{D/\Gamma} = H_g \cup H_{g-1}\cup\cdots \cup H_1\cup H_0\) coincides with the Satake compactification.
Consider now the example of \(F_{2d}\). Recall that \(D\) is a component of \(\Omega \mathrel{\vcenter{:}}=\left\{{z\in {\mathbf{P}}L_{\mathbf{C}}{~\mathrel{\Big\vert}~}z^2 = 0,\, z\overline{z} > 0}\right\}\) and \(D {}^{ \vee }= \left\{{z\in {\mathbf{P}}L_{\mathbf{C}}{~\mathrel{\Big\vert}~}z^2 = 0}\right\}\) is a quadric. Stratify \(\partial D = \partial_1 D {\textstyle\coprod}\partial_0 D\), noting that \(\partial_0 D = D {}^{ \vee }\cap{\mathbf{P}}L_{\mathbf{R}}\). Then all points of \(\partial_0 D\) are of the form \({\mathbf{P}}(\left\langle{v}\right\rangle_{\mathbf{C}})\) where \(v\in L_{\mathbf{R}}, v^2 = 0\) is isotropic. All components in \(\partial_1 D\) are of the form \({\mathbf{P}}(\left\langle{v, w}\right\rangle_{\mathbf{C}}) \cap\partial_1 D\) where \(\left\langle{v, w}\right\rangle_{\mathbf{R}}\) varies in \({\operatorname{OGr}}_2(L_{\mathbf{R}})\). Restricting to rational components, one considers \({\operatorname{OGr}}_2(L)\) instead, i.e. subspaces \(E_{\mathbf{R}}\) arising from sublattices \(E \leq L\). We thus obtain bijections \[\begin{align*} B_0(D) &\rightleftharpoons{\operatorname{OGr}}_1(L) \\ {\mathbf{P}}\left\langle{v}\right\rangle_{\mathbf{C}}&\rightleftharpoons\left\langle{v}\right\rangle_{\mathbf{Z}} \end{align*}\] and \[\begin{align*} B_1(D) &\rightleftharpoons{\operatorname{OGr}}_2(L) \\ {\mathbf{P}}\left\langle{v, w}\right\rangle_{\mathbf{C}}\cap\partial_1 D &\rightleftharpoons\left\langle{v, w}\right\rangle_{\mathbf{Z}} \end{align*}\]
One can then write \[\begin{align*} \overline{D/\Gamma} = D/\Gamma \cup \coprod_{[\left\langle{v}\right\rangle_{\mathbf{Z}}] \in {\operatorname{OGr}}_1(L)/\Gamma } p_v \cup \coprod_{[\left\langle{v, w}\right\rangle_{\mathbf{Z}}] \in {\operatorname{OGr}}_2(L)/\Gamma } {{\mathbf{P}}\left\langle{v, w}\right\rangle_{\mathbf{C}}\cap\partial_1 D \over N_\Gamma(\left\langle{v, w}\right\rangle_{\mathbf{Z}})} \end{align*}\] where \(N_\Gamma(E) = {\operatorname{Stab}}_\Gamma(E) / \mathrm{Fix}_\Gamma(E)\) can be identified with the image of \({\operatorname{Stab}}_\Gamma(E)\) in \({\operatorname{SL}}(E) \cong {\operatorname{SL}}_2({\mathbf{Z}})\).