Hartogs-Bochner type theorem in projective space (Q1426915)

A Hartogs-Bochner type theorem for real hypersurfaces in the projective space \(P_n(\mathbb C)\) (\(n\geq 2\)) is proved; namely: if \(M\) is a \(\mathcal C^2\) hypersurface that divides \(P_n(\mathbb C)\) into two connected open sets \(\Omega_1\), \(\Omega_2\) and has at most one open CR orbit, then there is an index \(i\in\{1,2\}\) such that all CR functions of class \(\mathcal C^k\) on \(M\), with \(k\geq 1\), holomorphically extend to holomorphic functions on \(\Omega_i\) which are \(\mathcal C^k\) up to the boundary. This result considerably extends various results on the Hartogs-Bochner phenomenon in projective spaces that were known in the literature.