Minimal kernels and compact analytic objects in complex surfaces

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 Mongodi et al. (Indiana Univ. Math. J., 67(2), 899-935 (2018); Int. J. Math., 28(8), 1750063, 16 (2017))) and we present some new results which stem from this somehow more local (or less global) viewpoint (see Sections 4.2, 4.3, and 5).