Graded rings of Hermitian modular forms with singularities (Q2686598)

In the paper under the review, the authors study rings of meromorphic modular forms with poles supported on certain rational quadratic divisors, and provide a number of examples of free algebras of modular forms. For \(d \in \{ 4, 7, 8, 11, 15, 19, 20, 24 \}\), let \(K = \mathbb{Q}(\sqrt{-d})\) denote the imaginary quadratic-field and let \(\mathcal{O}_K\) stand for its ring of integers. The authors prove that there is a Heegner divisor \(H_d\) such that the ring of symmetric meromorphic modular forms for the group \(\mathrm{SO}_{2,2}(\mathcal{O}_K)\) is freely generated. Let \(\mathbf{H}_2\) denote the Hermitian upper half-space of degree two, and let us denote by \(\Gamma_K\) the group generated by \(\mathrm{SO}_{2,2}(\mathcal{O}_K)\) and a reflection \(\sigma\) such that the modular forms on \(\Gamma_K\) are exactly those symmetric modular forms on \(\mathrm{SO}_{2,2}(\mathcal{O}_K)\). It is shown that the Looijenga compactification of \(\Gamma_K \setminus (\mathbf{H}_2 - H_d)\) is a complex weighted projective space of dimension four.