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. 
Dror Varolin