I know a version of Hartogs' Theorem in the book *An introduction to complex analysis*(p 30) by Hormander, namely
Hartogs' Theorem when K compact with complement being simply connected

I also have learned another version in *Hodge Theory and Complex Algebraic Geometry I* (p34) by Voisin. Hartogs' Theorem when $K=\{z_1=z_2=0\}$

My question: are there any generalizations of Hartogs' Theorem, with respect to $K$ (i.e. we only change the condition of $K$, confining the stage in $\mathbb{C}^{n}$ )?

Any clue is welcome. Many Thanks!