Big line bundles on unitary modular varieties

Abstract

We prove that unitary modular varieties are of general type if their dimension n ą 196 or the discriminant of the imaginary quadratic field is sufficiently large, under the assumption that there exists at least one non-zero cusp form of low weight and special unitary groups are principal. This follows from the result that the line bundle, whose section is Hermitian modular forms vanishing on branch divisors, on unitary modular varieties is big, through the calculation of the Hirzebruch-Mumford volume. In particular, for Hermitian lattices whose determinant is odd square-free, we find that the associated special unitary groups are principal and there are only finitely many ones whose corresponding varieties are not of general type, under the existence of cusp forms. Consequently, we formulate and partially show the finiteness of the number of Hermitian lattices admitting Hermitian reflective modular forms, which is a unitary analog of the conjecture proposed by Gritsenko-Nikulin for quadratic forms. Our study is motivated by the celebrated work of Tai, Freitag, and Mumford on Siegel modular varieties and of Kondō, Gritsenko-Hulek-Sankaran, and Ma on orthogonal modular varieties.