  We show that under very general assumptions the partial Bergman kernel
function of sections vanishing along an analytic hypersurface has exponential
decay in a neighborhood of the vanishing locus. Considering an ample line
bundle, we obtain a uniform estimate of the Bergman kernel function associated
to a singular metric along the hypersurface. Finally, we study the asymptotics
of the partial Bergman kernel function on a given compact set and near the
vanishing locus.
