Let $X$ be a projective Riemann surface, and denote with $\mathscr O_X$ the sheaf of regular functions (the strucure sheaf) and with $\mathscr O_{X^{an}}$ the sheaf of holomorphic functions on $X$. Clearly $\mathscr O_X\subset\mathscr O_{X^{an}} $.
An Hermitian invertible sheaf $\mathscr L$ on $X$ is a locally free sheaf of rank $1$ such that for each point $x\in X$ there is an Hermitian metric $h_x$ on the complex vector space $L(x):=\mathscr L_x\otimes_{\mathscr O_x} \mathbb C$. Moreover these metrics $h_x$ vary smoothly on $X$.
Now my question is:
is $\mathscr L$ a sheaf of $\mathscr O_X$-modules or of $\mathscr O_{X^{an}}$-modules? Of course if the answer is the latter, then the definition of $L(x)$ has to be changed as well.
It seems that books adopt the convention that $\mathscr L$ is a $\mathscr O_X$-module (see for example Miranda's book), but there is a problem with this. We loose the equivalence between hermitian invertible sheaf and hermitian line bundles! Indeed we are considering only line bundles with regular transition functions.