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

The celebrated Ohsawa--Takegoshi extension theorem for $L^2$ holomorphic functions on bounded pseudoconvex domains in $\mathbb C^n$ is a fundamental result in several complex variables and complex geometry. Ohsawa conjectured in 1995 that the same theorem still holds for more general bounded complete K\"ahler domains in $\mathbb C^n$. Recently, Chen--Wu--Wang confirmed this conjecture in a special case. In this paper we extend their result to the case of holomorphic sections of twisted canonical bundles over relatively compact complete K\"{a}hler domains in Stein manifolds. As an application we prove a Hartogs type extension theorem for plurisubharmonic functions across a compact complete pluripolar set, which is complementary to a classical result of Shiffman and can be seen as an analogue of the Skoda--El Mir extension theorem for plurisubharmonic functions -- a result that has been vacant since at least 1985.