Degenerations of Calabi-Yau threefolds and BCOV invariants

In [M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279 304; M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311-427], by expressing the physical quantity F-1 in two distinct ways, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable equivalence between Ray-Singer analytic torsion and elliptic instanton numbers for Calabi-Yau threefolds. After their discovery, in [H. Fang, Z. Lu and K.-I. Yoshikawa, Analytic torsion for Calabi-Yau threefolds, J. Differential Geom. 80 (2008) 175-250], a holomorphic torsion invariant for Calabi-Yau threefolds corresponding to F-1, called BCOV invariant, was constructed. In this article, we study the asymptotic behavior of BCOV invariants for algebraic one-parameter degenerations of Calabi-Yau threefolds. We prove the rationality of the coefficient of logarithmic divergence and give its geometric expression by using a semi-stable reduction of the given family.