Modular forms with poles on hyperplane arrangements

We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated with root lattices. We give a uniform construction of 147 hyperplane arrangements on type IV symmetric domains for which the algebras of modular forms with constrained poles are free and therefore the Looijenga compactifications of the arrangement complements are weighted projective spaces. We also construct eight free algebras of modular forms on complex balls with poles on hyperplane arrangements. The most striking example is the discriminant kernel of the 2U D 11 lattice, which admits a free algebra on 14 meromorphic generators. Along the way, we determine minimal systems of generators for non-free algebras of orthogonal modular forms for 26 reducible root lattices and prove the modularity of formal Fourier-Jacobi series associated with them. By exploiting an identity between weight 1 singular additive and multiplicative lifts on 2U D11, 11 , we prove that the additive lift of any (possibly weak) theta block of positive weight and q-order 1 is a Borcherds product. The special case of holomorphic theta blocks of one elliptic variable is the theta block conjecture of Gritsenko, Poor and Yuen.