The Borcherds–Dolgachev family of modular forms on symmetric tube domains: constructing the analogues of the Enriques $\Phi$-function for the remaining quaternionic, exceptional, real, and complex cases

Borcherds (1996) constructed an automorphic form $\Phi$ of weight 4 on the 10-dimensional symmetric tube domain attached to the even lattice $M=L\oplus II_{1,1}(2)$, $L=E_8(-2)\oplus II_{1,1}$, given as a twisted denominator function of the fake monster Lie superalgebra, and used it to prove that the moduli space of complex Enriques surfaces is an affine variety with a copy of the affine line removed. Dolgachev observed that $\Phi$ should be one case of a family: for each real division algebra $R\in\{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$ and each $n$, the tube domain $S_n(R)+iS_n(R)^+$ of Hermitian $n\times n$ matrices over $R$ carries a natural arithmetic group $\Gamma(R)$, and there should exist a $\Gamma(R)$-modular form whose nonvanishing locus is the period space of a family of Calabi–Yau $n$-folds; at the time, existence was known only for $n=2$, $R\neq\mathbb{H}$. The machinery questions surrounding $\Phi$ (an automorphic product theory beyond level 1, the circular proof that $\Phi$ has no extra zeros, and the indirect proof of full automorphy through a proper subgroup $\Gamma_3$) have all been resolved or made obsolete by the modern Borcherds-lift theory and by independent constructions of $\Phi$ (Kondō; Freitag–Salvati Manni; Yoshikawa; Kawaguchi–Mukai–Yoshikawa). The geometric conjecture for $R=\mathbb{C}$, $n\ge 3$ (Dolgachev's double covers of $\mathbb{P}^n$ branched along $2n+2$ hyperplanes) was disproved by Sheng–Xu–Zuo (2015), who proved the relevant monodromy groups Zariski dense. The form-existence question for the remaining symmetric cases — quaternionic $R=\mathbb{H}$ (all $n\ge 2$), the 27-dimensional exceptional domain ($R=\mathbb{O}$, $n=3$), and $R=\mathbb{R}$ or $\mathbb{C}$ with $n\ge 3$ — remains open: the adjacent literatures (quaternionic and exceptional modular forms, reflective modular forms) stop short of constructing any such reflective product form. The problem asks for the construction of these forms, with the Enriques-type properties expected of an analogue of $\Phi$ (divisor supported on reflective rational hyperplanes, automorphy under the full group with character), or a proof that no such form exists for stated cases.

Background

An Enriques surface is a smooth complex projective surface whose universal cover is a K3 surface (a simply connected complex surface carrying a nowhere-vanishing holomorphic 2-form), modulo a fixed-point-free holomorphic involution. The local Torelli theorem (which says that the period of the holomorphic 2-form locally determines the surface) and global period theory realize the moduli of K3 and Enriques surfaces as quotients of hermitian symmetric domains (complex domains on which a Lie group acts transitively with compact stabilizer) by arithmetic groups of lattice automorphisms. In Borcherds's 1996 paper the relevant lattice is $M=L\oplus II_{1,1}(2)$, where $L=E_8(-2)\oplus II_{1,1}$, $E_8$ is the even root lattice of the exceptional simple Lie algebra of that name, $II_{1,1}$ (also written $U$) is the even unimodular hyperbolic plane — the rank-2 lattice with basis $e,f$ and pairing $e\cdot f=1$; a lattice is called even when all its vector norms are even integers and unimodular when it equals its own dual — and $L(n)$ denotes the same lattice with its bilinear form multiplied by $n$, so $M$ is an even lattice of signature $(2,10)$ (two positive and ten negative directions of the form), isomorphic to $E_8(-2)\oplus U\oplus U(2)$. The hermitian symmetric space $\Omega$ is the set of lines in $M\otimes\mathbb{C}$ spanned by isotropic vectors $z$ (vectors of squared norm $(z,z)=0$) with $(z,\bar z)>0$; it is a 10-dimensional complex symmetric domain of type IV (one of the four infinite families of irreducible bounded symmetric domains in Cartan's classification), has two components, has a tube model $L\otimes\mathbb{R}+iC$ ($C$ a positive cone), and $O_M(\mathbb{Z})^+$ denotes the index-2 subgroup of the integer automorphism group of $M$ preserving the components. The moduli space of Enriques surfaces is the quotient of $\Omega$ by $O_M(\mathbb{Z})^+$ with the divisor of norm $-2$ vectors removed. Here an automorphic form is a holomorphic function on the domain that transforms under each group element by a specific factor of automorphy, whose exponent is the weight of the form, and whose residual phase defines a character. The paper constructs an automorphic form $\Phi$ for $O_M(\mathbb{Z})^+$ of weight 4 with a quadratic character $\chi$ (a character with values in $\{\pm1\}$; reflections in norm $-2$ vectors map to $-1$), arising as a twisted denominator function of the fake monster Lie superalgebra (the fake monster Lie algebra is a certain generalized Kac–Moody algebra — an infinite-dimensional Lie algebra described by an infinite root system — constructed from the Leech lattice, the unique even unimodular positive-definite lattice of rank 24; its twisted denominator function is an infinite product, here taken with respect to an order-2 automorphism of the Leech lattice fixing an 8-dimensional subspace). The key property is that the zeros of $\Phi$ are exactly the hyperplanes of norm $-2$ vectors of $M$, with multiplicity 1, so that $\Phi^{-1}$ trivializes the fourth power of the ample line bundle over the Enriques moduli space and shows it is affine with a copy of the affine line removed. A Borcherds product (automorphic product) is an automorphic form obtained as such a twisted denominator function; in modern language it is the Borcherds lift of a vector-valued modular form transforming under the Weil representation of the lattice (a natural action of $SL_2(\mathbb{Z})$ on the discriminant group of the lattice, the finite quotient of the dual lattice by the lattice itself), whose principal part (the polar part of its Fourier expansion) prescribes the divisor. The level-1 theory available in 1995 — level 1 meaning that it applied only to unimodular lattices, whose discriminant group is trivial — did not cover the lattice $M$ (whose discriminant group is a nontrivial 2-group, so the lattice has level 2), and the paper therefore proved the automorphy of $\Phi$ by an ad hoc analytic argument. In the introduction the paper then records Dolgachev's suggestion of an infinite family containing $\Phi$. Let $R$ be one of the four real division algebras $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ (real numbers, complex numbers, Hamilton quaternions, Cayley octonions), $S_n(R)$ the Hermitian $n\times n$ matrices over $R$, and $S_n(R)^+$ the cone of positive definite ones. The tube domain $S_n(R)+iS_n(R)^+$ is a symmetric bounded domain except when $R=\mathbb{O}$ and $n>3$. For $n=2$, $R=\mathbb{O}$ it is the domain $\Omega$ above. The arithmetic groups are $\Gamma(R)=Sp(2n,\mathbb{Z})$ for $R=\mathbb{R}$ (the Siegel modular group), $GL(2n,\mathbb{Z}[i])$ for $R=\mathbb{C}$ (the Hermitian modular group over the Gaussian integers), $SU(2n,\mathcal{O}_\mathbb{H})$ for $R=\mathbb{H}$ ($\mathcal{O}_\mathbb{H}$ is the Hurwitz order, the ring of integral quaternions, and $SU(2n,\mathcal{O}_\mathbb{H})$ is the quaternionic modular group), and $O_M(\mathbb{Z})$ for $R=\mathbb{O}$, $n=2$. The paper states: there might be a similar $\Gamma$-modular form on each of these spaces, the complement of its zeroes should be the period space of a family of (possibly non-simply-connected) Calabi–Yau $n$-folds, and existence is known for $n=2$, $R\neq\mathbb{H}$: for $R=\mathbb{R}$ the domain is the genus-2 Siegel space (the moduli space of principally polarized complex abelian surfaces) and the family is Kummer surfaces (quotients of abelian surfaces by $\pm1$); for $R=\mathbb{C}$ the domain is 4-dimensional of type $I_{2,2}\cong IV_4$ (Cartan types of bounded symmetric domains) and the family is K3 surfaces that are resolutions of double covers of $\mathbb{P}^2$ branched over six lines, the corresponding form being due to Matsumoto; for $R=\mathbb{O}$ it is $\Phi$ itself (the case $R=\mathbb{O}$, $n=3$ is the 27-dimensional exceptional symmetric domain of type $E_7$, studied by Krieg's school). Dolgachev conjectured that for $R=\mathbb{C}$ and arbitrary $n$ the family is the resolution of double covers of $\mathbb{P}^n$ branched along $2n+2$ hyperplanes in general position. A form analogous to $\Phi$ would in particular be a reflective modular form: an automorphic form whose divisor is a union of rational reflective hyperplanes (hyperplanes fixed pointwise by reflections of the arithmetic group and defined over the rationals), the shape characteristic of denominator functions of generalized Kac–Moody algebras. It is also expected to be of singular weight, that is, of the smallest weight in which nonzero holomorphic modular forms on the domain in question can exist (for the signature-$(2,10)$ Enriques domain this minimal weight is 4, the weight of $\Phi$).

Problem Statement

For the symmetric tube domains $S_n(R)+iS_n(R)^+$ over the four real division algebras $R\in\{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$ with their natural arithmetic groups $\Gamma(R)$ ($Sp(2n,\mathbb{Z})$ for $\mathbb{R}$; $GL(2n,\mathbb{Z}[i])$ for $\mathbb{C}$; $SU(2n,\mathcal{O}_\mathbb{H})$ for $\mathbb{H}$, with $\mathcal{O}_\mathbb{H}$ the Hurwitz order; the orthogonal group $O_M(\mathbb{Z})$ of the Enriques lattice for $(\mathbb{O},2)$, and its natural analogue for $(\mathbb{O},3)$, where identifying the correct arithmetic group on the exceptional domain is itself part of the case), the question is whether there exists on each space a $\Gamma$-modular form analogous to the Enriques form $\Phi$: an automorphic form whose zero locus is a union of rational reflective hyperplanes; in the geometrically meaningful cases the complement of its zeros is the period space of a family of Calabi–Yau $n$-folds. Existence is known exactly for $n=2$, $R\neq\mathbb{H}$ (Siegel genus 2, Matsumoto's form, and $\Phi$), and the case $R=\mathbb{O}$, $n>3$ is out of scope because the domain is not symmetric. The still-open problem is: for each remaining symmetric case — (i) $R=\mathbb{H}$, any $n\ge 2$ (first case $n=2$, a 6-dimensional domain); (ii) $R=\mathbb{O}$, $n=3$ (the 27-dimensional exceptional domain of type $E_7$); (iii) $R=\mathbb{R}$, $n\ge 3$; (iv) $R=\mathbb{C}$, $n\ge 3$ — construct such a $\Gamma(R)$-modular form with divisor supported on reflective rational hyperplanes, automorphic under the full group with a character, or prove that no such form exists for a stated case. The geometric part of the question — interpreting the complement of the zeros as a Calabi–Yau period space, including Dolgachev's conjecture on double covers of $\mathbb{P}^n$ branched along $2n+2$ hyperplanes — survives only where not already refuted: it was disproved for $R=\mathbb{C}$, $n\ge 3$, and remains meaningful as part of the analogy for the quaternionic, exceptional, and real cases if a form is found. The family question concerns all four remaining cases.

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