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 Kähler 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 Kähler 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.