One version of Hartogs' extension theorem is the following (see, e.g. [1], Theorem 5B, p. 50).

Theorem. Let $U \subset \mathbb{C}^n$ be open and let $X \subset U$ be a complex-analytic subvariety of codimension $>1$. Then, any holomorphic function $U \setminus X \to \mathbb{C}$ has a unique holomorphic extension $U \to \mathbb{C}$.

Question. Is this theorem still true if we replace $X$ by a real-analytic subvariety of codimension $>2$?

Note that I am still considering holomorphic functions. (The real-analytic version of Hartogs' theorem is false.)


[1] Whitney, Hassler. Complex analytic varieties. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. xii+399 pp.


1 Answer 1


The answer to your question is yes. It is enough to have the $2n-2$ dimensional Hausdorff measure of $X$ be zero and $X$ is closed. See the book of E. M. Chirka Complex Analytic Sets, page 298 proposition 3.


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