  We use classical homological algebra methods to get results on the Hartogs
extension phenomenon for holomorphic sections of holomorphic vector bundles
over complex analytic varieties. Namely, we study behaviours of the Hartogs
extension property with respect to the open embeddings, proper maps,
compactifications, and relations with the compact supports first cohomology and
with properties of boundaries of complex analytic varieties. As an application,
we get a convex-geometric criterion of the Hartogs phenomenon for complex
almost homogeneous algebraic $G$-varieties, where $G$ is a semiabelian variety.
