Motivic integration and the birational invariance of BCOV invariants

Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called the BCOV invariant. The BCOV invariant is conjecturally related to the Gromov-Witten theory via mirror symmetry. Based upon previous work of the second author, we prove the conjecture that birational Calabi-Yau manifolds have the same BCOV invariant. We also extend the construction of the BCOV invariant to Calabi-Yau varieties with Kawamata log terminal singularities, and prove its birational invariance for Calabi-Yau varieties with canonical singularities. We provide an interpretation of our construction using the theory of motivic integration.