Space of germs of holomorphic functions

The space $\mathcal{G}$ of germs over a domain $D$ has basic open sheetsfor holomorphic $f$ on open $U\subseteq D$. The projection $\pi ([f{]}_z)=z$ restricts to a homeomorphism on each sheet, and these projections form holomorphic coordinate charts.

The projection $[f{]}_z\mapsto z$ is a local biholomorphism for the sheet topology on the space of germs of holomorphic functions. It need not be a topological covering map: branches may end at singularities even when another branch remains over that base point.

The inverse germs of $w^3-3w$ over $\mathbb{C}$ form the connected surface $\mathbb{C}\setminus \{-1,1\}$. Its projection is surjective and locally biholomorphic, but the fibres have three points generically and one point at each critical value $\pm 2$, so it is not an evenly covered topological covering.

The evaluation map $\mathcal{E}([f{]}_z)=f(z)$ is holomorphic. On the sheet associated with $(U,f)$, its coordinate expression is exactly the holomorphic function $f$.

Over $D=\mathbb{C}\setminus \{z:z^8=1\}$, the two-valued square root is represented byThe maps $\pi (z,w)=z$ and $\mathcal{E}(z,w)=w$ identify this unbranched double cover analytically with the corresponding component of the space of germs.