Hartogs and open embeddings, proper maps, compactifications, cohomologies

In these (not-completed) notes, we study the Hartogs extension phenomenon for holomorphic sections of holomorphic vector bundles over complex analytic varieties. Namely, we study properties of the Hartogs extension phenomenon with respect to the open embeddings, proper maps, compactifications, relations with the compact supports first cohomology, and the Lefschetz type property for sections of sheaves. 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.