Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac--Moody algebras

TL;DR

This paper presents six series of lattice-polarized K3 surfaces with automorphic discriminant, linking Lorentzian Kac--Moody algebras and arithmetic mirror symmetry to provide explicit examples.

Contribution

It introduces new explicit examples of lattice-polarized K3 surfaces with automorphic discriminant based on Lorentzian Kac--Moody algebras and arithmetic mirror symmetry.

Findings

Six series of lattice-polarized K3 surfaces with automorphic discriminant

Connections established between K3 surfaces, Kac--Moody algebras, and mirror symmetry

Explicit examples illustrating theoretical concepts

Abstract

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant.

Tables

Table 1. Table 1: S 𝑆 S -polarized K3 surfaces with automorphic discriminant.

$S^{mir}$	$T=$	weight	$S={\left(T\right)}_{3U\oplus 2E_8}^{⟂}$	$q_S$
	$U\left(m\right)\oplus S^{mir}$	of $\Phi \left(z\right)$
$U\oplus A_1$	$m=1$	$35$	$U\oplus E_8\oplus E_7$	$2_1^{+1}$
$U\oplus 2A_1$	$m=1$	$34$	$U\oplus E_8\oplus D_6$	$2_2^{+2}$
$U\oplus A_2$	$m=1$	$45$	$U\oplus E_8\oplus E_6$	$3^{-1}$
$U\oplus 3A_1$	$m=1$	$33$	$U\oplus E_7\oplus D_6$	$2_3^{+3}$
$U\oplus A_3$	$m=1$	$54$	$U\oplus E_8\oplus D_5$	$4_3^{-1}$
$U\oplus 4A_1$	$m=1$	$32$	$U\oplus D_6\oplus D_6$	$2_4^{+4}$
$U\oplus 2A_2$	$m=1$	$42$	$U\oplus E_6\oplus E_6$	$3^{+2}$
$U\oplus A_4$	$m=1$	$62$	$U\oplus E_8\oplus A_4$	$5^{+1}$
$U\oplus D_4$	$m=1$	$72$	$U\oplus E_8\oplus D_4$	$2_{II}^{-2}$
$U\oplus D_4$	$m=2$	$40$	$U\left(2\right)\oplus E_8\oplus D_4$	$2_{II}^{-4}$
$U\oplus A_5$	$m=1$	$69$	$U\oplus E_8\oplus A_2\oplus A_1$	$2_{-1}^{+1},3^{+1}$
$U\oplus D_5$	$m=1$	$88$	$U\oplus E_8\oplus A_3$	$4_5^{-1}$
$U\oplus 3A_2$	$m=1$	$39$	$U\oplus E_6\oplus 2A_2$	$3^{-3}$
$U\oplus 2A_3$	$m=1$	$48$	$U\oplus 2D_5$	$4_6^{+2}$
$U\oplus A_6$	$m=1$	$75$		$7^{-1}$
$U\oplus D_6$	$m=1$	$102$	$U\oplus E_8\oplus 2A_1$	$2_{-2}^{+2}$
$U\oplus E_6$	$m=1$	$120$	$U\oplus E_8\oplus A_2$	$3^{+1}$
$U\oplus A_7$	$m=1$	$80$	$U\oplus E_8\oplus ⟨-8⟩$	$8_{-1}^{+1}$
$U\oplus D_7$	$m=1$	$114$	$U\oplus E_8\oplus ⟨-4⟩$	$4_{-1}^{+1}$
$U\oplus E_7$	$m=1$	$165$	$U\oplus E_8\oplus A_1$	$2_{-1}^{+1}$
$U\oplus 2D_4$	$m=1$	$60$	$U\oplus 2D_4$	$2_{II}^{+4}$
$U\oplus D_8$	$m=1$	$124$	$U\oplus D_8$	$2_{II}^{+2}$
$U\oplus E_8$	$m=1$	$252$	$U\oplus E_8$	$0$
$U\left(2\right)\oplus 2D_4$	$m=1$	$28$	$U\left(2\right)\oplus 2D_4$	$2_{II}^{+6}$
$U\oplus 2E_8$	$m=1$	$132$	$U$	$0$

Table 2. Table 2: S 𝑆 S -polarized K3 surfaces with automorphic discriminant.

$S^{mir}$	$T=$	weight	$S={\left(T\right)}_{3U\oplus 2E_8}^{⟂}$	$q_S$
	$U\left(m\right)\oplus S^{mir}$	of $\Phi \left(z\right)$
$U$	$m=1$	$12$	$U\oplus E_8\oplus E_8$	$0$
$U\oplus A_1\left(2\right)$	$m=1$	$12$	$U\oplus E_8\oplus D_7$	$4_1^{+1}$
$U\oplus A_1\left(3\right)$	$m=1$	$12$	$U\oplus E_8\oplus E_6\oplus A_1$	$2_{-1}^{+1},3^{-1}$
$U\oplus A_1\left(4\right)$	$m=1$	$12$	$U\oplus E_8\oplus A_7$	$8_1^{+1}$
$U\oplus 2⟨-4⟩$	$m=1$	$12$	$U\oplus D_7\oplus D_7$	$4_2^{+2}$
$U\oplus A_2\left(2\right)$	$m=1$	$12$	$U\oplus E_8\oplus D_4\oplus A_2$	$2_{II}^{-2},3^{+1}$
$U\oplus A_2\left(3\right)$	$m=1$	$12$	$U\oplus E_8\oplus {\left(A_2\left(3\right)\right)}_{E_8}^{⟂}$	$3^{-1},9^{-1}$
$U\oplus A_3\left(2\right)$	$m=1$	$12$	$U\oplus E_8\oplus {\left(A_3\left(2\right)\right)}_{E_8}^{⟂}$	$2_{II}^{-2},8_3^{-1}$
$U\oplus D_4\left(2\right)$	$m=1$	$12$	$U\oplus E_8\oplus D_4\left(2\right)$	$2_{II}^{-2},4_{II}^{-2}$
$U\oplus E_8\left(2\right)$	$m=1$	$12$	$U\oplus E_8\left(2\right)$	$2_{II}^{+8}$

Table 3. Table 3: S 𝑆 S -polarized K3 surfaces with automorphic discriminant.

$S^{mir}$	$T=$	weight	$S={\left(T\right)}_{3U\oplus 2E_8}^{⟂}$	$q_S$
	$U\left(m\right)\oplus S^{mir}$	of $\Phi \left(z\right)$
$⟨2⟩\oplus A_1$	$m=2$	$12$	$U\left(2\right)\oplus E_8\oplus E_7\oplus A_1$	$2_0^{+4}$
$⟨2⟩\oplus 2A_1$	$m=2$	$11$	$U\left(2\right)\oplus E_7\oplus E_7\oplus A_1$	$2_1^{+5}$
$⟨2⟩\oplus 3A_1$	$m=2$	$10$	$U\left(2\right)\oplus E_7\oplus D_6\oplus A_1$	$2_2^{+6}$
$⟨2⟩\oplus 4A_1$	$m=2$	$9$	$U\left(2\right)\oplus D_6\oplus D_6\oplus A_1$	$2_3^{+7}$
$⟨2⟩\oplus 5A_1$	$m=2$	$8$	$U\oplus D_6\oplus 6A_1$	$2_4^{+8}$
$⟨2⟩\oplus 6A_1$	$m=2$	$7$	$U\left(2\right)\oplus D_6\oplus 5A_1$	$2_5^{+9}$
$⟨2⟩\oplus 7A_1$	$m=2$	$6$	$U\left(2\right)\oplus D_4\oplus 6A_1$	$2_6^{+10}$
$⟨2⟩\oplus 8A_1$	$m=2$	$5$	$U\left(2\right)\oplus E_8\left(2\right)\oplus A_1$	$2_7^{+11}$

Table 4. Table 4: S 𝑆 S -polarized K3 surfaces with automorphic discriminant.

$S^{mir}$	$T=$	weight	$S={\left(T\right)}_{3U\oplus 2E_8}^{⟂}$	$q_S$
	$U\left(m\right)\oplus S^{mir}$	of $\Phi \left(z\right)$
$U\left(2\right)\oplus D_4$	$m=1$	$40$	$U\left(2\right)\oplus E_8\oplus D_4$	$2_{II}^{-4}$
$U\left(2\right)\oplus D_4$	$m=2$	$24$	$U\oplus 3D_4$	$2_{II}^{-6}$
$U\left(4\right)\oplus D_4$	$m=4$	$6$	$U\left(4\right)\oplus {\left(U\left(4\right)\oplus D_4\right)}_{U\oplus 2E_8}^{⟂}$	$2_{II}^{-2},4_{II}^{+4}$

Table 5. Table 5: S 𝑆 S -polarized K3 surfaces with automorphic discriminant.

$S^{mir}$	$T=U\left(m\right)$	weight	$S={\left(T\right)}_{3U\oplus 2E_8}^{⟂}$	$q_S$
	$\oplus S^{mir}$	of $\Phi \left(z\right)$
$U\left(4\right)\oplus A_1$	$m=4$	$5$	$U\left(4\right)\oplus {\left(U\left(4\right)\right)}_{U\oplus E_8}^{⟂}\oplus E_7$	$2_1^{+1},4_{II}^{+4}$
$U\left(4\right)\oplus 2A_1$	$m=4$	$4$	$U\left(4\right)\oplus {\left(U\left(4\right)\right)}_{U\oplus E_8}^{⟂}\oplus D_6$	$2_2^{+2},4_{II}^{+4}$
$U\left(4\right)\oplus 3A_1$	$m=4$	$3$	$U\left(4\right)\oplus {\left(U\left(4\right)\right)}_{U\oplus E_8}^{⟂}\oplus$	$2_3^{+3},4_{II}^{+4}$
			$\oplus D_4\oplus A_1$
$U\left(4\right)\oplus 4A_1$	$m=4$	$2$	$U\left(4\right)\oplus {\left(U\left(4\right)\right)}_{U\oplus E_8}^{⟂}\oplus 4A_1$	$2_4^{+4},4_{II}^{+4}$

Table 6. Table 6: S 𝑆 S -polarized K3 surfaces with automorphic discriminant.

$S^{mir}$	$T=U\left(m\right)$	weight	$S={\left(T\right)}_{3U\oplus 2E_8}^{⟂}$	$q_S$
	$\oplus S^{mir}$	of $\Phi \left(z\right)$
$U\left(3\right)\oplus A_2$	$m=3$	$9$	$U\left(3\right)\oplus {\left(U\left(3\right)\right)}_{U\oplus E_8}^{⟂}\oplus E_6$	$3^{-5}$
$U\left(3\right)\oplus 2A_2$	$m=3$	$6$	$U\left(3\right)\oplus {\left(U\left(3\right)\right)}_{U\oplus E_8}^{⟂}\oplus 2A_2$	$3^{+6}$
$U\left(3\right)\oplus 3A_2$	$m=3$	$3$	$U\left(3\right)\oplus {\left(U\left(3\right)\oplus A_2\right)}_{U\oplus E_8}^{⟂}\oplus 2A_2$	$3^{-7}$

Equations

$$V(S)=\{x\in S\otimes \mathbb{R}|x^2>0\}$$

$$S_X=H^2(X,\mathbb{Z})\cap H^{1,1}(X)=\{x\in H^2(X,\mathbb{Z})|x\cdot {\omega }_X=0\}\subset H^2(X,\mathbb{Z})$$

$$H^{2,0}(X)=\mathbb{C}{\omega }_X\subset T_X\otimes \mathbb{C}\subset T\otimes \mathbb{C}$$

$$\Omega (T)=\{\mathbb{C}\omega \subset T\otimes \mathbb{C}|\omega \cdot \omega =0and\omega \cdot \overline{\omega }>0{\}}^{+}$$

$$\mathrm{Discr}(T)=\bigcup _{\beta \in T^{(2)}}{D}_{\beta },$$

$$D_{\beta }=\{\mathbb{C}\omega \in \Omega (T)|\omega \cdot \beta =0\}$$

$$O^{+}(T)=\{g\in O(T)|g(\Omega (T))=\Omega (T)\}$$

$$Mod(S)=\bigcup _{1\le k\le n}{G}_k\backslash (\Omega (T_k)-\mathrm{Discr}(T_k))$$

$${\tilde{O}}^{+}(T)=\{g\in O^{+}(T)|g{|}_{T^{\ast }/T}=\mathrm{id}\}$$

$$U=\left\langle\begin{array}[]{cc}0&1\\ 1&0\end{array}\right\rangle$$

$$\{F_k=0\}=\sum _r{m}_r{D}_r$$

$$O(U(2)\oplus (⟨2⟩\oplus (k+1)⟨-2⟩))\cong O(U\oplus (⟨1⟩\oplus (k+1)⟨-1⟩))\cong O(U\oplus U\oplus D_k)$$

Full text

Dedicated to É.B. Vinberg on the occasion of his 80th Birthday

1 Introduction

Using results of our recent paper [12] and our previouse papers, we construct many even hyperbolic lattices $S$ such that $S$-polarized complex K3 surfaces $X$ have an automorphic discriminant.

We recall that for a $S$-polarized K3 surface $X$ a primitive embedding $S\subset S_{X}$ is fixed where $S_{X}$ is the Picard lattice of $X$. We say that such $X$ is degenerate (or it belongs to the discriminant) if there exists $\delta\in(S)^{\perp}_{S_{X}}$ such that $\delta^{2}=-2$. By geometry of K3 surfaces, it then follows that $X$ has no a polarization $h$ from $S$. By Global Torelli Theorem [23] and epimorphicity of period map for K3 surfaces [15], moduli of such K3 surfaces are covered by the corresponding hermitian symmetric domains, and algebraic functions on moduli are the corresponding automorphic forms on these domains. A holomorphic automorphic form is called discriminant if the support of its zero divisor is equal to the preimage of the discriminant of moduli of such K3 surfaces. If a discriminant automorphic form exists, the discriminant is then called automorphic.

For example, for $S=\mathbb{Z}h$ of the rank one with $h^{2}=n$ where $n\geq 2$ is even (that is for usual polarized K3 surfaces), it is well-known that the discriminant automorphic form exists for $n=2$. Borcherds constructed the discriminant automorphic form for $n=2$ explicitly (see [2, pp. 200–201]). It was shown in [22] that for infinite number of even $n\geq 2$ the discriminant automorphic form does not exist (probably, it was the first result in this direction). Later, Looijenga [16] showed that the discriminant automorphic form does not exist and the discriminant is not automorphic for all $n>2$.

Here, we ask about examples of automorphic discriminants for $\operatorname{rk}S\geq 2$. See some finiteness results in Ma [17].

In Sect. 2, we give necessary definitions about $S$-polarized K3 surfaces and their discriminants and automorphic discriminants.

In Sect. 3, we formulate the main Theorems 3.1 and 3.2 which give six series of even hyperbolic lattices $S$ of $\operatorname{rk}S\geq 2$ such that $S$-polarized K3 surfaces have an automorphic discriminant. They are given in Tables 1—6. All these examples are related to the Lorentzian Kac-Moody algebras constructed in [12], those are hyperbolic automorphic Kac–Moody (super) Lie algebras. The corresponding discriminant automorphic forms are given in [12] and define such Kac–Moody algebras $\mathfrak{g}$, and give their denominator identities.

It would be interesting to understand geometric meaning of these automorphic forms and Kac–Moody algebras for the geometry of the corresponding K3 surfaces. For example, we know that if the weight of the discriminant automorphic form is larger than the dimension of the moduli space then the moduli space is at least uniruled (see Theorem 3.4 in §3).

2 Lattice-polarized K3 surfaces and their

moduli and discriminants

We refer to [19] about lattices. We recall that a lattice $M$ (equivalently, a non-degenerate integral symmetric bilinear form) means that $M$ is a free $\mathbb{Z}$-module $M$ of a finite rank with symmetric $\mathbb{Z}$-bilinear non-degenerate pairing $x\cdot y\in\mathbb{Z}$ for $x,\,y\in M$. By signature of $M$, we mean the signature of the corresponding real form $M\otimes\mathbb{R}$ over $\mathbb{R}$ (that is the numbers $(t_{(+)},\,t_{(-)})$ of positive and negative squares respectively). A lattice $M$ of the signature $(1,\operatorname{rk}M-1)$ is called hyperbolic. A lattice $M$ is called even if $x^{2}=x\cdot x$ is even for any $x\in M$. By $O(M)$, we denote the automorphism group of a lattice $M$. Each element $\delta\in M$ with $0\not=\delta^{2}$ and $\delta^{2}|2(\delta\cdot M)$ (it is called root) defines the reflection $s_{\delta}:x\mapsto x-[(2(x\cdot\delta)/\delta^{2}]\delta$ for $x\in M$. Evidently, $s_{\delta}\in O(M)$, $s_{\delta}(\delta)=-\delta$ and $s_{\delta}$ is identical on $\delta^{\perp}_{M}$. By $W^{(2)}(M)\subset O(M)$, we denote the subgroup generated by reflections in all elements $\delta\in M$ with $\delta^{2}=-2$ (they are all roots).

Let $S$ be a hyperbolic lattice. Let

[TABLE]

be the cone of $S$. It has two connected components $V^{+}(S)$ and $V^{-}(S)=-V^{+}(S)$. We fix one of them, $V^{+}(S)$, and the corresponding hyperbolic space $\mathcal{L}(S)=V^{+}(S)/\mathbb{R}_{++}$. Here $\mathbb{R}_{++}$ denotes all positive real numbers and $\mathbb{R}_{+}$ denotes all non-negative real numbers. Let ${\rm Amp}(S)$, ${\rm Amp}(S)/\mathbb{R}_{++}$ be interiors of fundamental chambers for the reflection group $W^{(2)}(S)$ in $V^{+}(S)$ and $\mathcal{L}(S)$ respectively. We fix one of them. Thus, we fix the pair $(V^{+}(S),{\rm Amp}(S))$. They are defined uniquely up to the action of $O(S)$. We call the pair as the ample cone of $S$. It is equivalent to ${\rm Amp}(S)$ or ${\rm Amp}(S)/\mathbb{R}_{++}$.

Let $X$ be a Kälerian K3 surface (for example, see [23], [15], [3], [25], [24] about such surfaces), that is $X$ is a non-singular compact complex surface with trivial canonical class $K_{X}$ (equivalently, $0\not=\omega_{X}\in H^{2,0}(X)=\Omega^{2}[X]$ has the zero divisor) and such that the irregularity $q(X)$ is equal to [math] (equivalently, $X$ has no non-zero holomorphic $1$-dimensional holomorphic forms). Then $H^{2,0}(X)=\mathbb{C}\omega_{X}$ and $H^{2}(X,\mathbb{Z})$ with the intersection pairing is an even unimodular (that is with the determinant $\pm 1$) lattice $L_{K3}$ of signature $(3,19)$. The primitive sublattice

[TABLE]

is the Picard lattice of $X$ generated by the first Chern classes of all line bundles over $X$. Here primitive means that $H^{2}(X,\mathbb{Z})/S_{X}$ has no torsion. By the difinition, $S_{X}$ can be either negative definite, semi-negative definite, or hyperbolic lattice. By Kodaira, the last case is exactly the case when $X$ is projective algebraic.

Further, we assume that $X$ is algebraic. We denote by $V^{+}(S_{X})=V(X)$ the half cone of $S_{X}$ which contains a polarization of $X$, and by ${\rm Amp}(X)\subset V(X)$ the ample cone of $X$. Then ${\rm Amp}(S_{X})={\rm Amp}(X)$ gives an ample cone of $S_{X}$, see [23].

Further, we fix an even hyperbolic lattice $S$ and its ample cone ${\rm Amp}(S)$.

We remind to a reader (e. g. see [6], [5] and [18]) that a $S$-polarized K3 surface $X$ means that a primitive embedding $S\subset S_{X}$ of lattices is fixed such that ${\rm Amp}(S)\cap{\rm Amp}(X)\not=\emptyset$.

If instead of the last condition only the conditions ${\rm Amp}(S)\cap\overline{{\rm Amp}(X)}\not=\emptyset$ and ${\rm Amp}(S)\cap{\rm Amp}(X)=\emptyset$ are satisfied, then we say that $X$* is a degenerate $S$-polarized K3 surface, equivalently, $X$ belongs to the discriminant of moduli of $S$-polarized K3 surfaces.* By geometry of K3 surfaces (see [23]), it happens only if there exists $\delta\in(S)^{\perp}_{S_{X}}$ such that $\delta^{2}=-2$.

By global Torelli Theorem for K3 surfaces [23] and empimorphicity of period map for K3 surfaces [15], for general $S$-polarized K3 surfaces we have $S_{X}=S$ and ${\rm Amp}(X)={\rm Amp}(S)$, for non-degenerate $S$-polarized K3 surfaces $X$, the $(S)^{\perp}_{S_{X}}$ has no elements $\delta$ with $\delta^{2}=-2$ and ${\rm Amp}(X)\cap{\rm Amp}(S)\not=\emptyset$, and for degenerate $S$-polarized K3 surfaces $X$, the $(S)^{\perp}_{S_{X}}$ has elements $\delta$ with $\delta^{2}=-2$ and only ${\rm Amp}(S)\cap\overline{{\rm Amp}(X)}\not=\emptyset$ is valid, equivalently, $X$ belongs to the discriminant of moduli of $S$-polarized K3 surfaces.

For a $S$-polarized K3 surface $X$, let us consider periods

[TABLE]

where $T_{X}=(S_{X})^{\perp}_{H^{2}(X,\mathbb{Z})}$ is the transcendental lattice of $X$ and $T=(S)^{\perp}_{H^{2}(X,\mathbb{Z})}$ is the transcendental lattice of the $S$-polarization. The periods give a point in IV type Hermitian symmetric domain

[TABLE]

where $+$ means a choice of one of two connected components. This point belongs to the complement of the discriminant

[TABLE]

where

[TABLE]

is the rational quadratic divisor which is orthogonal to $\beta\in T$ with $\beta^{2}<0$; we recall that $\beta^{2}=-2$ for $\beta\in T^{(2)}$. Of course, $D_{\beta}=D_{-\beta}$, and we identify $\pm\beta$ in this definition. Further,

[TABLE]

is the group of automorphisms of $T$ which preserve the connected component $\Omega(T)$.

By considering all possible isomorphism classes $T_{1},\dots,T_{n}$ of the transcendental lattice $T$ for all primitive embeddings $S\subset L_{K3}$, we correspond to a $S$-polarized K3 surface $X$ a point in

[TABLE]

where $G_{k}\subset O^{+}(T_{k})$ is an appropriate finite index subgroup. By global Torelli Theorem [23] and epimorphicity of the period map for K3 surfaces [15], each point of $Mod(S)$ corresponds to some $S$-polarized K3 surface $X$.

We recall that a holomorphic function $\Phi$ on $\Omega(T)$ is called an automorphic form of a weight $d\in\mathbb{N}$ if $\Phi$ is homogeneous of the degree $(-d)$ and it is symmetric with respect to a subgroup $H\subset O^{+}(T)$ of finite index.

Finally, we can give a defintion:

Definition 2.1**.**

We fix an even hyperbolic lattice $S$. We say that $S$-polarized K3 surfaces have an automorphic discriminant if for each $1\leq k\leq n$ in (2.2) there exists a holomorphic automorphic form on $\Omega(T_{k})$ such that the support of its zero divisor is equal to $Discr(T_{k})$ in (2.1). Then we call this automorphic form discriminant automorphic form.

The stable orthogonal group

[TABLE]

is a subgroup of finite index of $O^{+}(T)$. For a primitive embedding $S\subset L_{K3}$ and $T=(S)^{\perp}_{L_{K3}}$, the group $\widetilde{O}^{+}(T)$ consists of automorphisms from $O^{+}(T)$ which can be continued to an element of $O(L_{K3})$ identically on $S$. Thus, we can assume that $\widetilde{O}^{+}(T_{k})\subset G_{k}$.

3 Lattice-polarized K3 surfaces with

automorphic discriminant related to

Lorentzian Kac–Moody algebras with Weyl groups of $2$-reflections

Below, we use the following notations for lattices. We use $\oplus$ for the orthogonal sum of lattices. By $tM$, we denote the orthogonal sum of $t$ copies of $M$. By $A_{k}$, $k\geq 1$, $D_{m}$, $m\geq 4$, $E_{l}$, $l=6,7,8$, we denote the standard root lattices with Dynkin diagrams $\mathbb{A}_{k}$, $\mathbb{D}_{m}$, $\mathbb{E}_{l}$ respectively and the roots with square $(-2)$. For a lattice $M$, we denote by $M(t)$ the lattice which is obtained from $M$ by multiplication by $0\not=t\in\mathbb{Q}$ of the bilinear form of the lattice $M$ if the form of $M(t)$ remains integral. By $\langle A\rangle$, we denote the lattice with the symmetric matrix $A$. We denote by

[TABLE]

the even unimodular lattice of signature $(1,1)$. For example, $L_{K3}\cong 3U\oplus 2E_{8}$.

We remind to a reader that for an integer lattice $M$ we have the canonical embedding $M\subset M^{\ast}=Hom(M,\mathbb{Z})$. It defines a (finite) discrminant group $A_{M}=M^{\ast}/M$. By continuing the symmetric bilinear form of the lattice $M$ to $M^{\ast}$, we obtain a finite symmetric bilinear form $b_{M}$ on $A_{M}$ with values in $\mathbb{Q}/\mathbb{Z}$ and a finite quadratic form $q_{M}$ on $A_{M}$ with values in $\mathbb{Q}/2\mathbb{Z}$, if $M$ is even. They are called the discriminant forms of the lattice $M$.

If there are no other conditions, by $(M)^{\perp}_{L}$ we mean an orthogonal complement to a lattice $M$ in a lattice $L$ for some primitive embedding $M\subset L$. For the most cases of the Theorems 3.1 and 3.2 below, the orthogonal complement is unique up to isomorphism. For other case, it does not matter which isomorphism class we shall take.

We have the following six series of examples of even hyperbolic lattices $S$ of $\operatorname{rk}S\geq 2$ such that $S$-polarized K3 surfaces have an automorphic discriminant. They are given in the theorems below.

Theorem 3.1**.**

For the hyperbolic lattices $S$ which are given in the last columns of the Tables 1—6 below, $S$-polarized K3 surfaces have an automorphic discriminant. We also give the discriminant quadratic form $q_{S}$ of $S$ in notations of [4]. The even hyperbolic lattice $S$ is defined by its rank and $q_{S}$ uniquely up to isomorphism (see proofs below).

For all these cases, the transcendental lattice $T=(S)^{\perp}_{L_{K3}}$, $L_{K3}=3U\oplus 2E_{8}$, is unique up to isomorphism, and its isomorphism class is equal to $T=U(m)\oplus S^{mir}$ where the hyperbolic lattice $S^{mir}$ is shown in the first column and $m$ is shown in the second column of the table in the same line as $S$.

Theorem 3.2**.**

For all cases of Theorem 3.1, the discriminant automorphic form $\Phi(z)$ has the Fourier expansion with integral coefficients at the zero dimensional cusp defined by the decomposition $T=U(m)\oplus S^{mir}$ (see [12]), $z\in S^{mir}\otimes\mathbb{R}+\sqrt{-1}\,V^{+}(S^{mir})$. The Fourier coefficients define a Lorentzian (hyperbolic and automorphic) Kac–Moody super-algebra $\mathfrak{g}$ which is graded by the hyperbolic lattice $S^{mir}$. The $\Phi(z)$ has an infinite product (Borcherds) expansion which gives multiplicities of roots of this algebra. See [13], [14], [1], [2].

The divisor of $\Phi(z)$ is sum of rational quadratic divisors $D_{\alpha}$, $\alpha\in T^{(2)}$, with multiplicities one.

The $S^{mir}$-polarized K3 surfaces can be considered as mirror symmetric to $S$-polarized K3 surfaces by mirror symmetry considered in [6], [5], [10], [11]. They have the remarkable property that there exists $\rho\in S^{mir}\otimes\mathbb{Q}$ such that $\rho\cdot E=1$ for each non-singular rational curve $E\subset X$ with $S_{X}=S^{mir}$ (for $\rho^{2}>0$ and $\operatorname{rk}S^{mir}=4$, such $S^{mir}$ are in the list of $14$ lattices which were found by É.B. Vinberg in [26]; about other $S^{mir}$ see [20] and [21]).

Proof.

Theorems 3.1 and 3.2 are mainly reformulations of the results of [12] using the discriminants forms technique for integer lattices which was developed in [19].

Let $S$ be a lattice of one of Tables 1—6. By results of [19], we have $T=(S)^{\perp}_{3U\oplus 2E_{8}}\cong U(m)\oplus S^{mir}$ where $S^{mir}$ and $m$ are shown in the same line of the table as $S$. Here, it is important that the discriminant quadratic forms $q_{T}$ and $q_{S}$ are related as $q_{T}\cong-q_{S}$ since $T\perp S$ in the unimodular lattice $3U\oplus 2E_{8}$. Vice a versa, $(S)^{\perp}_{3U\oplus 2E_{8}}=T$ and $(T)^{\perp}_{3U\oplus 2E_{8}}=S$ for some primitive ebmeddings $S\subset 3U\oplus 2E_{8}$ and $T\subset 3U\oplus 2E_{8}$ if signatures of $T$, $S$ and $3U\oplus 2E_{8}$ agree and $q_{T}\cong-q_{S}$; the signature $(t_{(+)},t_{(-)})$ together with the discriminant quadratic form $q$ define the genus of an even lattice; Theorem [19, Theorem 1.13.1] (which uses results by M. Kneser) gives conditions when an even indefinite lattice with the invariants $(t_{(+)},t_{(-)},q)$ is unique up to isomorphism.

For all $S^{mir}$ and $m$ which are shown in lines of Tables 1—6, the automorphic form $\Phi(z)$ with the properties mentioned in the Theorems 3.1 and 3.2 is constructed in [12]. For the lattices of Table 1, it is done in [12, Theorem 4.3 and Proposition 4.1]; of Table 2, in [12, Theorem 4.4]; of Table 3, in [12, Theorem 6.1]; of Table 4, in [12, Theorems 6.2 and 6.3]; of Table 5, in [12, Lemma 6.4]; of Table 6, in [12, Theorem 6.5].

By results of [19] which were mentioned above, we have $S=(T)^{\perp}_{3U\oplus E_{8}}$ is unique up to isomorphism, and $S$ is shown in the Tables.

These considerations give the proof. ∎

In many cases, existence of the automorphic discriminant tells us that the moduli space of the corresponding $S$-polarized K3 surfaces has a special geometry. The following criterion is valid.

Theorem 3.3**.**

(See [8, Theorem 2.1].) Let $\Omega(T)$ be a connected component of the type $IV$ domain associated to a lattice $T$ of signature $(2,n)$ with $n\geq 3$ and let $\Gamma\subset\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(T)$ be an arithmetic subgroup of finite index of the orthogonal group. Let $\widetilde{B}=\sum_{r}D_{r}$ in $\Omega(T)$ be the divisorial part of the ramification locus of the quotient map $\Omega(T)\to\Gamma\backslash\Omega(T)$. (This means that the reflection $s_{r}$ or $-s_{r}$ belongs to $\Gamma$). Assume that a modular form $F_{k}$ with respect to $\Gamma$ of weight $k$ with a (finite order) character exists, such that

[TABLE]

where the $m_{r}$ are non-negative integers. Let $m=\max\{m_{r}\}$ (which must be $>0$ by Koecher’s principle). If $k>m\cdot n$, then $\Gamma^{\prime}\backslash\mathcal{D}$ is uniruled for every arithmetic group $\Gamma^{\prime}$ containing $\Gamma$.

Using this criterion, we prove

Theorem 3.4**.**

The moduli space of $S$-polarized K3 surfaces is at least uniruled if $S$ is any lattice of Table $1$ and Table $2$, a lattice from the first five lines of Table $3$ (till the lattice ${\langle{2}\rangle}\oplus 5A_{1}$), the first two lines of Tables $4$ and the first line of Tables $5$ and $6$.

Proof.

The moduli space of $S$-polarized K3 surfaces is defined in (2.2). For any lattice $S$ in Tables $1$–$6$ there is only one isomorphism class of the corresponding lattices $T$, i.e. there is only one term in (2.2). The modular group $G=G_{1}$ of the moduli space always contains the stable orthogonal group $\widetilde{O}^{+}(T)$ acting trivially on the discriminant quadratic form of $T$. The divisor $D_{r}$ with $r^{2}=-2$, $r\in T$, always belongs to the ramification divisor since $s_{r}\in\widetilde{O}^{+}(T)$. We note that $\widetilde{O}^{+}(T)$ is generated by $-2$-reflections for the most part of the lattices from Tables $1$ and $2$ (see [9]). By construction (see [12, §4]), any discriminant automorphic form from Tables $1$ and $2$ is a modular form with respect to $\widetilde{O}^{+}(T)$ with character det with the simplest possible divisor ${\rm Discr}(T)$ of multiplicity one. The weight of the discriminant automorphic form is shown in the Tables. If the dimension $n$ of the moduli space is larger than $2$, we apply Theorem 3.3. If $n=1$ or $2$, the corresponding modular varieties are at least unirational.

The construction of the discriminant automorphic forms of Table 3 uses the isomorphism

[TABLE]

(see [12, Lemma 6.1]). Moreover, the reflections with respect to $-2$-vectors of ${\langle{2}\rangle}\oplus(k+1){\langle{-2}\rangle}$ correspond to the reflections with respect to $-4$-reflective vectors of $U\oplus D_{k}$ or $-1$-vectors of $U\oplus D_{k}^{*}$. If $k\neq 4$, then all $-1$-reflective vectors of $2U\oplus D_{k}^{*}$ belong to the unique \widetilde{\mathop{\hbox{}\mathrm{O}}}\nolimits^{+}(2U\oplus D_{k})-orbit which is equal to the set of $-1$-vectors in $2U\oplus k{\langle{-1}\rangle}$. If $k=4$, then there are three such \widetilde{\mathop{\hbox{}\mathrm{O}}}\nolimits^{+}(2U\oplus D_{4})-orbits, and one of them coincides with the $-1$-vectors in $2U\oplus k{\langle{-1}\rangle}$.

The discriminant automorphic forms of Table $3$ (see [12, §6]) are modular with respect to the full orthogonal group $O^{+}(2U\oplus D_{k})$ if $k\neq 4$ and with a subgroup $\widetilde{O}^{+}(2U\oplus D_{4})$ containing $\widetilde{O}^{+}(U(2)\oplus({\langle{2}\rangle}\oplus(5{\langle{-2}\rangle}))$. If $k\leq 5$, then the weight of the discriminant automorphic form is strictly larger than the dimension of the moduli space.

The similar argument works for the remaining cases with the modular forms constructed in [12, §6.3–6.5]. ∎

Remark. In each Table $3$–$6$, there exists one discriminant automorphic form with weight which is equal to the dimension of the homogeneous domain. It follows that the Kodaira dimension of a finite quotient of the corresponding moduli space is equal to [math]. (See a criterion in [7] and [8, Theorem 1.3].) We shall consider these cases in some details later.