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The celebrated Ohsawa-Takegoshi extension theorem for $L^2$ holomorphicfunctions on bounded pseudoconvex domains in $\mathbb C^n$ is an important andpowerful tool in several complex variables and complex geometry. Ohsawaconjectured in 1995 that the same theorem holds for more general boundedcomplete K\"ahler domains in $\mathbb C^n$. Recently, Chen-Wu-Wang confirmedthis conjecture in a special case. In this paper we extend their result to thecase of holomorphic sections of twisted canonical bundles over relativelycompact complete K\"{a}hler domains in Stein manifolds. As an application weprove a Hartogs type extension theorem for plurisubharmonic functions across acompact complete pluripolar set, which is complementary to a classical theoremof Shiffman and can be thought of as an analogue of the Skoda-El Mir extensiontheorem for plurisubharmonic functions.

On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions