We introduce a notion of partial geometric invariants which naturally satisfies the gluing law and hence is particularly useful in studying contributions from singularity. Holomorphic torsion is the complex version of the Ray-Singer analytic torsion for Riemannian manifolds. Unlike the Ray-Singer analytic torsion, it does not satisfy any gluing law. As an application we prove that the holomorphic torsion of log-Enriques surfaces is given by the Petersson norm of the Borcherds modular form. The last part is joint work with Ken-Ichi Yoshikawa.