The Levi problem on projective manifolds (Q1896778)

Let \(M\) be a compact complex manifold and \(E\) be a positive holomorphic line bundle on \(M\). If \(D\) is a given locally pseudoconvex domain in \(M\), we show that \(D\) is holomorphically convex with respect to global holomorphic sections of high enough tensor power \(E^r\) of \(E\). Hörmander's \(L^2\) estimates for the \(\overline {\partial}\)-operator are basic tools (precisely the adapted version for ideals of holomorphic sections obtained by Skoda) to construct global sections of \(E^r\) over \(D\) which blow up at a fixed boundary point of \(D\).