I am reading "Complex Geometry: an introduction", by D. Huybrechts.
In the definition, 1.1.22 says that a germ $X\subset \mathbb{C}^n$ in $0$ is called analytic if $X$ and $Z(f_1,f_2,\ldots,f_n)$ define the same germ, where $f_i \in \mathcal{O}_{\mathbb{C}^n,0}$. What is that mean ?
I understand that $Z(f_1,f_2,\ldots,f_n)\subset \mathbb{C}^n$ since it contains all the points where the functions $f_i$ vanishes. I don't understand how $Z(f_1,f_2,\ldots,f_n)$ is a germ and how a germ $X$ can be thought as a subset of $\mathbb{C}^n$.
The book doesn't define very rigorously what a germ is but from the definition 1.1.14 I assume it means it's the set of all the equivalence classes $(U,f)$. Where $(U,f)\sim (V,g)$ if there exist $W\subset U\cap V$ such that $f|_W=g|_W.$
Maybe I understand wrong the definition but can someone explain what we mean when we write the germ $X\subset \mathbb{C}^n$ ?
That is the germ of a function $f$. The germ of a subset $X \subset \mathbb C^n$ at zero is an equivalence class of subsets. Two subsets $X, Y \subset \mathbb C^n$ define the same germ at zero, if there exists a neighbourhood $U$ of $0 \in \mathbb C^n$, such that $$X \cap U = Y \cap U.$$ So $X$ and $Y$ look the same near zero.