  To prove that a modular variety is of general type, there are three types of
obstructions: reflective, cusp and elliptic obstructions. In this paper, we
give a quantitative estimate of the reflective obstructions for the unitary
case. This shows in particular that the reflective obstructions are small
enough in higher dimension, say greater than $138$. Our result reduces the
study of the Kodaira dimension of unitary modular varieties to the construction
of a cusp form of small weight in a quantitative manner. As a byproduct, we
formulate and partially prove the finiteness of Hermitian lattices admitting
reflective modular forms, which is a unitary analog of the conjecture by
Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective
obstructions uses Prasad's volume formula.
