V Extensions of holomorphic and CR functions on manifolds

The aim of this chapter is to study the Hartogs–Bochner phenomenon on complex analytic manifolds. We start by studying the relationship between Hartogs’ phenomenon and the vanishing of the Dolbeault cohomology group with compact support in bidegree (0, 1). We then give some cohomological conditions which enable us to extend a CR function of class $$C^\infty$$ defined on a subset of the boundary of a domain to a holomorphic function on the whole domain. This work generalises the geometric situation studied at the end of Chapter IV. This work generalises the geometric situation studied at the end of Chapter IV. Proving similar results for CR functions of class Ck requires two extra elements: a theorem on local resolutions of $$\partial$$ and an isomorphism theorem between the various cohomology groups $$H^{p,q}_{\alpha}$$ (X). This isomorphism theorem follows from the local resolution and some sheaf-theoretic results which are given in Appendix B. The existence of the resolution is proved by solving $$\overline{\partial}$$ in convex domains with C2 boundary using a new integral formula, the Cauchy–Fantappié formula.