  Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for
Calabi-Yau manifolds, which is called the BCOV invariant. In this paper, we
extend the BCOV invariant to such pairs $(X,D)$, where $X$ is a compact
K\"ahler manifold and $D$ is a pluricanonical divisor on X with simple normal
crossing support. We also study the behavior of the extended BCOV invariant
under blow-up. The results in this paper lead to a joint work with Fu proving
that birational Calabi-Yau manifolds have the same BCOV invariant.
