Does the maximum modulus principle hold for general analytic sets?

Question

Let $U \subset \mathbb C^n$ be a domain and $X \subset U$ a connected analytic subset. Does the maximum modulus principle hold for $X$? That is, if I have a holomorphic function $f : U \to \mathbb C$ whose modulus attains a maximum value on $X$, is $f$ then constant on $X$?

(Obviously, this is true if $X$ does not have singularities, because then $X$ is itself a complex manifold. But what if $X$ does have singularities?)

Answer 1

Yes, the maximum modulus principle holds for $X$. See Gunning and Rossi, “Analytic Functions of Several Complex Variables”, chapter III, theorem B.16, p. 106:

16. Theorem. Let $V$ be a connected subvariety of the domain $U$ in $\mathbb C^n$. If $f$ is holomorphic in $U$, and its modulus attains a maximum in $V$, then $f$ is constant on $V$.