Several complex variables and analytic spaces Extension phenomena and analytic continuation Proposition

Zeros of holomorphic functions are not isolated

Builds on Holomorphic function of several variables Hartogs's extension theorem

Statement

Let $n \ge 2$ , let $D \subseteq \C^n$ be an open set, and let $f$ be a holomorphic function on $D$ . If $p \in D$ and $f(p) = 0$ , then every neighbourhood of $p$ contains a point $q \ne p$ with $f(q) = 0$ .

Remarks

Suppose instead that $p$ is the only zero of $f$ in a small open ball $B$ around $p$ . Then $1/f$ is holomorphic on $B \setminus \set{p}$ , which is connected. By Hartogs's extension theorem with $K = \set{p}$ , it extends to a holomorphic function $g$ on $B$ . Then $g f = 1$ on $B \setminus \set{p}$ , hence at $p$ by continuity, which contradicts $f(p) = 0$ .

Example: $f(z_1, z_2) = z_1^2 + z_2^2$ vanishes at the origin, and its zeros near the origin form the two complex lines $z_1 = i z_2$ and $z_1 = -i z_2$ .

The hypothesis $n \ge 2$ is necessary: $f(z) = z$ on $\C$ has the isolated zero $0$ .

Going deeper

For a student

The zero set of a holomorphic function that is not identically zero is an analytic hypersurface, of complex dimension $n - 1$ at each of its points. This proposition is the first step toward that fact. Divisors in several variables are made of hypersurfaces instead.

Reading path

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Compact set concept
Connected set concept
Holomorphic function of several variables concept
Open set concept
Hartogs's extension theorem statement
Zeros of holomorphic functions are not isolated statement

Tag	`5EZRN3812E`
URI	`m2://5EZRN3812E`
Kind	statement, proposition
Writing	W0 stub
MSC	32A60, 32A10
Status	This copy of the site has no build data. Run `m2 status 5EZRN3812E`.
About	Holomorphic function of several variables
Uses	Holomorphic function of several variables, Hartogs's extension theorem