A Counterexample to Hartogs’ Type Extension of Holomorphic Line Bundles

Hartogs Type Extension Theorems

Let ∆ ⊆ C be the open unit disc and let Σ ⊆ ∆×∆ be a compact set such that K = Σ ∪ (∂∆×∆) is a connected set. It is a classical result by Hartogs that if Σ is an analytic variety over ∆ with the boundary in ∂∆×∆, then every function holomorphic in a connected neighbourhood of K extends holomorphically to a neighbourhood of ∆ × ∆. It is proved that the same conclusion holds if Σ is a ‘continuous...

A General Version of the Hartogs Extension Theorem for Separately Holomorphic Mappings between Complex Analytic Spaces

Using recent development in Poletsky theory of discs, we prove the following result: Let X, Y be two complex manifolds, let Z be a complex analytic space which possesses the Hartogs extension property, let A (resp. B) be a non locally pluripolar subset of X (resp. Y ). We show that every separately holomorphic mapping f : W := (A × Y ) ∪ (X × B) −→ Z extends to a holomorphic mapping f̂ on Ŵ := {...

Extension of holomorphic bundles to the disc (and Serre’s Problem on Stein bundles)

Para-Kahler tangent bundles of constant para-holomorphic sectional curvature

We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagona...

Some Remarks on Hartogs’ Extension Lemma

Motivated by a result and a question by E. M. Chirka we consider the Hartogs’ extension property for some connected sets in C2 of the form K = Σ ∪ (∂Δ × Δ), where Σ is a possibly nonconnected compact subset of Δ×Δ ⊂ C2.