BCOV invariant and blow-up

Bershadsky, Cecotti, Ooguri and Vafa constructed a real-valued invariant for Calabi-Yau manifolds, which is now called the BCOV invariant. In this paper, we extend the BCOV invariant to such pairs (X, D), where X is a compact Kahler 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-ups. The results in this paper lead to a joint work with Fu proving that birational Calabi-Yau manifolds have the same BCOV invariant.