Line bundle convexity of pseudoconvex domains in complex manifolds (Q749711)

If X is a complex manifold and E a holomorphic vector bundle on x with a hermitian metric, then X is called E-convex if to each infinite discrete set of points in X there is a holomorphic section of E over X which is unbounded on that set. We show that if U is a pseudoconvex domain with \(C^ 2\) boundary in a smooth projective variety M, and if L is a positive line bundle on M, then U is \(\otimes^ rL\)-convex for a large enough integer r. In the proof we construct a complete Kähler metric on U with controlled growth, showing that for domains with \(C^ 2\) boundary in a Kähler manifold, pseudoconvexity is equivalent to the existence of a complete Kähler metric. Unbounded local sections of \(\otimes^ rL\) are obtained using \({\bar \partial}\) methods for pseudoconvex domains in \({\mathbb{C}}^ n\). An \(L^ 2\)-cohomology argument is then used to construct unbounded global sections.