I am reading Deligne's "Equations differentielles à points réguliers singuliers" and got stuck on page 61 which translates as follows:
Let $X$ be a sparated complex analytic space, $Y$ a closed analytic subset of $X$ and $X^{*}:= X\backslash Y$ a complex manifold. Assume that $X$ can be imbedded into some $\mathbb{C}^{n}$. If $j_{i}:X\rightarrow \mathbb{C}^{n_{i}}$ $(i=1,2)$ are two such embeddings, then the Riemannian structures induced by the the ones on $\mathbb{C}^{n_{i}}$, denoted by $j_{i}^{*}g$ and $j_{2}^{*}g$ verify
For all compact $K$ in $X$ there are constants $A,B>0$ such that $$\begin{equation} j_{1}^{*}g \leq Ag_{2}^{*}g\leq Bg_{2}^{*}g\leq Bj_{1}^{*}g\qquad (*) \end{equation} $$ on $K\cap X^{*}$.
My intuition is that Deligne has made a typo there, since the symbols $g_{i}^{*}$ is not even defined. Can someone make something meaningful out of it?
One could try and replace (*) by $$\begin{equation} j_{1}^{*}g\leq A j_{2}^{*}g \text{ and }j_{2}^{*}g\leq Bj_{1}^{*}g \end{equation}$$ (inspired by the characterisation of the equivalence of two norms) but then there still lacks a notion of $\leq$-relation between Riemannian metrics. I do think that I could turn this into some meaningful definition but instead of guessing what Deligne might have thought while writing the notes I prefer to ask the community about it. Possibly this definition is standard and it is only I who do not know it.
