Marked M-polarized K3 data and m-admissible vector

Use the algebraic K3 convention: a K3 surface is a smooth connected compact complex surface that is simply connected and has trivial canonical line bundle. The K3 lattice $L$ is the even unimodular lattice of signature $(3,19)$, with its integral symmetric bilinear form. A marking is an isometry $\varphi :H^2(S;\mathbb{Z})\to L$.

Here a sublattice is primitive if its quotient in $L$ is torsion-free, and $\mathrm{Pic}(S)\subset H^2(S;\mathbb{Z})$ is the image of the first Chern class of holomorphic line bundles. An M-polarized marked K3 datum consists of a primitive sublattice $M\subset L$ of signature $(1,t)$, a marked K3 surface $(S,\varphi )$ with ${\varphi }^{-1}(M)\subseteq \mathrm{Pic}(S)$, and its orthogonal complement $T=M^{\perp }\subset L$. The corresponding period domain is $D_M=\{[\mathrm{\Omega }]\in \mathbb{P}(T{\otimes }_{\mathbb{Z}}\mathbb{C}):(\mathrm{\Omega },\mathrm{\Omega })=0,(\mathrm{\Omega },\bar{\mathrm{\Omega }})>0\}\mathrm{.}$

For this source, a Kähler class is a real cohomology class represented by a smooth closed positive real $(1,1)$-form; the intersection pairing is the K3 lattice pairing transported by $\varphi$. The root-hyperplane locus is ${\mathrm{\Delta }}_M=\{[\mathrm{\Omega }]\in D_M:\text{ some }0\mathrm{\ne }\alpha \in T\text{ has }(\alpha ,\mathrm{\Omega })=0\}$.

For a positive integer $m$, an isotropic vector $E\in T$ is m-admissible if there is an isotropic $E^{\mathrm{\prime }}\in T$ with $(E,E^{\mathrm{\prime }})=m$, and no $\alpha \in T$ satisfies either $0<(\alpha ,E)<m$ or $0<(\alpha ,E^{\mathrm{\prime }})<m$. The vector is isotropic when $(E,E)=0$. In this basis the rank-two lattice $U(m)$ has Gram matrix $\left(\begin{array}{cc}0& m\\ m& 0\end{array}\right)$; the source sets $\check{M}=(\mathbb{Z}E{)}_T^{\perp }\mathrm{/}\mathbb{Z}E$ and obtains $T\cong U(m)\oplus \check{M}$. “General period and Kähler class” below means $[\mathrm{\Omega }]\in D_M\setminus {\mathrm{\Delta }}_M$ and a Kähler class $\omega \in M\otimes \mathbb{R}$ with ${\omega }^{\perp }\cap M=0$. These are the exact lattice/genericity conventions used in Gross–Wilson §1; this definition asserts no elliptic fibration or special-Lagrangian fibration.

Compact Calabi–Yau manifold conventions