  In this paper, we want to study the link between the presence of compact
objects with some analytic structure and the global geometry of a weakly
complete surface. We begin with a brief survey of some now classic results on
the local geometry around a (complex) curve, which depends on the sign of its
self-intersection and, in the flat case, on some more refined invariants (see
the works of Grauert, Suzuki, Ueda). Then, we recall some results about the
propagation of compact curves and the existence of holomorphic functions (from
the works of Nishino and Ohsawa). With such considerations in mind, we give an
overview of the classification results for weakly complete surfaces that we
obtained in two joint papers with Slodkowski (see [MST18], [MST17] and we
present some new results which stem from this somehow more local (or less
global) viewpoint (see Sections 3.2, 3.3 and 4).
