  Period integrals arise from the comparison between Betti cohomologies and de
Rham cohomologies. In this paper, we give a formula for the determinant of
period integrals in terms of the canonical pairing between Picard group of rank
1 realizations and relative Chow group. It is compatible with the similar
formula for Galois modules proved by the first author, and as a consequence,
under the assumption of Gersten resolution, the determinant of the cohomology
of a determinantally motivic object is motivic.
