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 Kähler 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.
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