Admissible Hermitian metrics on families of line bundles over certain degenerating Riemann surfaces

Abstract

We show that a family of line bundles of degree zero over a plumbing family of Riemann surfaces with a separating (resp. non-separating) node p admits a nice (resp. almost nice) family of flat p-singular Hermitian metrics. As a consequence, we give necessary and sufficient conditions for a family of line bundles over such families of Riemann surfaces to admit an (almost) nice family of p-singular Hermitian metrics which are admissible with respect to the canonical/hyperbolic (1,1)-forms on the Riemann surfaces.