Graded rings of Hermitian modular forms with singularities

TL;DR

This paper investigates the structure of graded rings of meromorphic Hermitian modular forms of degree two, revealing they form polynomial algebras for certain imaginary quadratic fields and relating their compactifications to weighted projective spaces.

Contribution

It demonstrates that for specific imaginary quadratic fields, the graded rings are polynomial algebras without relations, and describes their compactifications as weighted projective spaces.

Findings

Graded rings form polynomial algebras for certain fields.

Looijenga compactifications are weighted projective spaces.

Poles supported on Heegner divisors characterize the forms.

Abstract

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $\mathrm{SU}_{2,2}(\mathcal{O}_K)$ where $K$ is the imaginary-quadratic number field of discriminant $-d$, $d \in \{4, 7, 8, 11, 15, 19, 20, 24\}$ we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.