# Minimal kernels and compact analytic objects in complex surfaces

## TL;DR

This paper explores the relationship between compact analytic objects and the global geometry of weakly complete complex surfaces, providing classification results and new insights from a local perspective.

## Contribution

It offers a comprehensive overview of existing results and introduces new findings on the local-global interplay in the geometry of weakly complete surfaces.

## Key findings

- Classification results for weakly complete surfaces
- Propagation of compact curves and holomorphic functions
- New local-global geometric insights

## Abstract

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).