Degenerations of Calabi–Yau threefolds and BCOV invariants

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  • Ken-Ichi Yoshikawa
    Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

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<jats:p> In [M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B405 (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<jats:sub>1</jats:sub> 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<jats:sub>1</jats:sub>, 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. </jats:p>

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