A Takayama-type extension theorem

We prove a theorem on the extension of holomorphic sections of powers of adjoint bundles from submanifolds of complex codimension 1 having non-trivial normal bundle. The first such result, due to Takayama, considers the case where the canonical bundle is twisted by a line bundle that is a sum of a big and nef line bundle and a ${\mathbb Q}$-divisor that has kawamata log terminal singularites on the submanifold from which extension occurs. In this paper we weaken the positivity assumptions on the twisting line bundle to what we believe to be the minimal positivity hypotheses. The main new idea is an $L^2$ extension theorem of Ohsawa-Takegoshi type, in which twisted canonical sections are extended from submanifolds with non-trivial normal bundle

Bergman interpolation on finite Riemann surfaces. Part I: Asymptotically Flat Case

We study the Bergman space interpolation problem of open Riemann surfaces obtained from a compact Riemann surface by removing a finite number of points. We equip such a surface with what we call an asymptotically flat conformal metric, i.e., a complete metric with zero curvature outside a compact subset. We then establish necessary and sufficient conditions for interpolation in weighted Bergman spaces over asymptotically flat Riemann surfaces.Comment: The main result has been corrected: Sequences of density <1 are still interpolating, but the density of an interpolation sequence is only shown to be at most 1. The corrected result is sharp, by work of Borichev-Lyubarskii. Also added a motivating section on Shapiro-Shields interpolation. Otherwise typos and minor errors corrected. To appear in Journal d'Analys

Stable manifolds of holomorphic diffeomorphisms

We consider stable manifolds of a holomorphic diffeomorphism of a complex manifold. Using a conjugation of the dynamics to a (non-stationary) polynomial normal form, we show that typical stable manifolds are biholomorphic to complex Euclidean space.Comment: 17 pages. Revised version. To appear in Inv. Mat