General Type of Orthogonal Modular Varieties

Definitions and mathematical statement

An even integral lattice $L$ is a finite-rank free abelian group with a nondegenerate integral symmetric bilinear form such that $(x,x)\in2{\mathbb{Z}}$ for every $x\in L$. Suppose its real signature is $(2,n)$. Its type-IV period domain is a connected component $\mathcal D_L$ of \[ \{[z]\in{\mathbb{P}}(L\otimes_{\mathbb{Z}}{\mathbb{C}}):(z,z)=0,\ (z,\overline z)>0\}. \] An associated orthogonal modular variety has the form $\Gamma_L\backslash\mathcal D_L$ for an appropriate arithmetic subgroup preserving that component. The record denotes its chosen modular variety by $\mathcal F(L)$, but does not specify $\Gamma_L$.

For that intended association $L\mapsto\mathcal F(L)$, the conjecture asks for a uniform integer $N$ such that every even integral lattice of signature $(2,n)$ with $n\geq N$ has $\mathcal F(L)$ of general type. The stronger stated prediction is \[ \forall n\geq36\ \forall L\text{ even integral of signature }(2,n), \qquad \mathcal F(L)\text{ is of general type}. \] Here general type is understood in the usual birational sense: a smooth projective model of a compactification has Kodaira dimension equal to $\dim\mathcal F(L)=n$. This is distinct from log general type. No unimodularity, discriminant bound, or special lattice family is imposed by the statement. The arithmetic group and compactification convention need source verification before the prediction can be treated as fully specified.

Short English statement

Ask for a dimension bound, uniform over every even integral lattice of signature $(2,n)$, ensuring that its orthogonal modular variety is of general type; the proposed bound is $n\geq36$. The exact modular group is missing.

Sources

• Local UnsolvedMath registry, permanent ID $30001016$, source code catalogue entry, "General Type of Orthogonal Modular Varieties." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Komplexe Analysis (2009), Oberwolfach Reports, pp. 2165--2218, DOI https://doi.org/10.4171/owr/2008/38.
• Supplied "Gritsenko, Hulek and Sankaran": https://arxiv.org/abs/math/0609774.
• Supplied "Ma, Kodaira dimension of orthogonal modular varieties": https://arxiv.org/abs/1701.03225.
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled partially solved; says general type is proved in important high-dimensional families, including the even unimodular case in Gritsenko--Hulek--Sankaran, but the sources found do not establish the stated uniform threshold for every even integral lattice. This is supplied dated curation, not independently verified.