2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165591Let $w$ be an Abelian differential on compact Riemann surface of genus $g\geq 1$. We obtain an explicit holomorphic factorization formula for $ζ$-regularized determinant of the Laplacian in flat conical metrics with trivial holonomy $|w|^2$, generalizing the classical Ray-Singer result in $g=1$.53 pages; the paper in present form is essentally self-contained; the proofs are improvedSpectral Theorytext