Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, III: relation with the BCOV invariant

A Calabi-Yau 4-fold of Camere-Garbagnati-Mongardi is a crepant resolution of the quotient of a hyperk\"ahler 4-fold by an antisymplectic involution. In this paper, we compare two different types of holomorphic torsion invariants; one is the BCOV invariant of the Calabi-Yau 4-fold of Camere-Garbagnati-Mongardi, and the other is the invariant of the corresponding $K3^{[2]}$-type manifold with involution introduced by the author in the preceding papers. As an application, in some special cases, we show that the BCOV invariant of those Calabi-Yau 4-folds is expressed as the Petersson norm of a certain Borcherds product.