In complex analytic geometry, metrics are a very well established tool. Let us first consider the case of the projective space $\mathrm{X}={\mathbf{P}}^{n}({\mathbf{C}})$ ; a point $x\in X$ is a $(n+1)$-tuple of homogeneous coordinates $[x_{0}:\dots:x_{n}]$, not all zero, and up to a scalar. Let $\pi\colon{\mathbf{C}}^{n+1}_{*}\rightarrow X$ be the canonical projection map, where the index $*$ means that we remove the origin $(0,\dots,0)$. The fibers of $\pi$ have a natural action of ${\mathbf{C}}^{*}$. The tautological line bundle $\mathscr{O}(1)$ has for sections $s$ over an open set $\mathrm{U}\subset{\mathbf{P}}^{n}({\mathbf{C}})$ the analytic functions $F_{s}$ on the open set $\pi^{-1}(\mathrm{U})\subset{\mathbf{C}}^{n+1}_{*}$ which are homogeneous of degree $1$. The Fubini-Study metric of $\mathscr{O}(1)$ assigns to the section $s$ the norm $\left\|{s}\right\|_{\mathrm{FS}}$ defined by

$$\left\|{s}\right\|_{{\mathrm{FS}}}([x_{0}:\dots:x_{n}])=\frac{\left|{F_{s}(x_{0},\dots,x_{n})}\right|}{\left(\left|{x_{0}}\right|^{2}+\dots+\left|{x_{n}}\right|^{2}\right)^{1/2}}.$$

It is more than continuous ; indeed, if $s$ is a local frame on an open set $\mathrm{U}$, then $\left\|{s}\right\|$ is a $\mathscr{C}^{\infty}$-function on $\mathrm{U}$ ; such metrics are called smooth.