# Hartogs' theorem for real-analytic subvarieties

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.)

References.

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

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.