K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space, II: A structure theorem for r(M) > 10

We study the structure of the invariant of 3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the 3 lattice with respect to the involution is strictly bigger than 10, we prove that this automorphic form is expressed as the tensor product of an explicit Borcherds lift and Igusa's Siegel modular form.