Irregular cusps of orthogonal modular varieties
Shouhei Ma
TL;DR
This work analyzes irregular cusps of orthogonal modular varieties, showing that the lattice for Fourier expansion can be strictly larger than the translation lattice at certain cusps, which affects cusp form vanishing and pluricanonical form construction. It develops a comprehensive framework for 0- and 1-dimensional cusps, including tube/Siegel domain models, stabilizers over $\mathbb{Q}$ and $\mathbb{Z}$, and toroidal compactifications, and introduces refined criteria for when irregular cusps influence Kodaira dimension calculations. A key outcome is a modified low slope cusp form criterion that accounts for irregular boundary divisors, ensuring correct general-type conclusions across various moduli spaces (e.g., K3s, O'Grady 10, generalized Kummer, special cubic fourfolds). The paper also provides explicit classifications and examples of irregular cusps in stable orthogonal groups, guiding practitioners in verifying general type results for families of orthogonal modular varieties. Overall, it clarifies when irregular cusps impact the relation between cusp forms and pluricanonical forms, and supplies practical tools for assessing Kodaira dimension in complex moduli problems.
Abstract
Irregular cusp of an orthogonal modular variety is a cusp where the lattice for Fourier expansion is strictly smaller than the lattice of translation. Presence of such a cusp affects the study of pluricanonical forms on the modular variety using modular forms. We study toroidal compactification over an irregular cusp, and clarify there the cusp form criterion for the calculation of Kodaira dimension. At the same time, we show that irregular cusps do not arise frequently: besides the cases when the group is neat or contains -1, we prove that the stable orthogonal groups of most (but not all) even lattices have no irregular cusp.