  In their study of genus-one string amplitude, Bershadsky-Cecotti-Ooguri-Vafa
discovered a remarkable identification between holomorphic Ray-Singer torsion
and instanton numbers for Calabi-Yau threefolds. The holomorphic torsion
invariant for Calabi-Yau threefolds corresponding to the genus-one string
amplitude is called BCOV invariant. In this paper, we establish an
identification between the BCOV invariants of Borcea-Voisin threefolds and
another holomorphic torsion invariants for K3 surfaces with involution. We also
introduce BCOV invariants for abelian Calabi-Yau orbifolds. Between
Borcea-Voisin orbifold and its crepant resolution, we compare their BCOV
invariants.
