Geometry of analytic continuation on complex manifolds – history, survey, and report

Abstract

Abstract Beginning with the state of art around 1953, solutions of the Levi problem on complex manifolds will be recalled at first up to Takayama’s result in 1998. Then, the activity of extending the results by the L 2 {L}^{2} method in these decades will be reported. The method is by exploiting the finite dimensionality of certain L 2 {L}^{2} ∂ ¯ \bar{\partial } -cohomology groups to prove that a Hermitian holomorphic line bundle L L over a complex manifold M M is bimeromorphically equivalent to an ample bundle when it is restricted to a bounded locally pseudoconvex domain Ω ⋐ M \Omega \hspace{0.15em}\Subset \hspace{0.15em}M under the positivity of L ∣ ∂ Ω {L| }_{\partial \Omega } and the regularity of ∂ Ω \partial \Omega .