After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which w... more After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa. Contents 1. Introduction 2. Calabi-Yau varieties with at most one ordinary double point 3. Quillen metrics 4. The BCOV invariant of Calabi-Yau manifolds 5. The singularity of the Quillen metric on the BCOV bundle 6. The cotangent sheaf of the Kuranishi space 7. Behaviors of the Weil-Petersson metric and the Hodge metric 8. The singularity of the BCOV invariant I -the case of ODP 9. The singularity of the BCOV invariant II -general degenerations 10. The curvature current of the BCOV invariant 11. The BCOV invariant of Calabi-Yau threefolds with h 1,2 = 1 12. The BCOV invariant of quintic mirror threefolds 13. The BCOV invariant of FHSV threefolds ANALYTIC TORSION FOR CALABI-YAU THREEFOLDS 5 an arbitrary Calabi-Yau manifold of arbitrary dimension, which we obtain using determinants of cohomologies [28], Quillen metrics [11], [44], and a Bott-Chern class like A(·). Then the BCOV Hermitian line of a Calabi-Yau manifold depends only on the complex structure of the manifold. The Hodge diamond of Calabi-Yau threefolds are so simple that the BCOV Hermitian line reduces to the scalar invariant τ BCOV in the case of threefolds. Hence Eq. (1.1) on P 1 \ D is deduced from the curvature formula for the BCOV Hermitian line bundles. (See Sect. 4).
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Papers by K. Yoshikawa