Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.


At that point, one states that one can describe a complex manifold $\mathcal{M}$, which as a set is in bijection with $M$.


My question is: The above statement is not really well-defined (it is only well-defined when given together with its proof); I would prefer a statement about existence of some complex manifold with some universal property. Could you specify such a property, or give a reference?



Moduli of stable bundles - analytic approach
$\mathcal{M}$ is what is called a coarse moduli space. In concrete terms, this means the following:


1) As a set, $\mathcal{M}$ can be viewed as the set of isomorphism classes of stable bundles (of rank $r$ and degree $d$):


2) Given any family of such vector bundles parametrized by an analytic space $S$ (that is, a vector bundle $\mathcal{E}$ on $S\times X$ such that for each $s\in S$, $\mathcal{E}\_s:=\mathcal{E}\_{|\{s\}\times X }$ is a stable bundle on $X$, of rank $r$ and degree $d$), the map $S\rightarrow \mathcal{M}$ given by $s\mapsto [\mathcal{E}\_s]$ is holomorphic.


It is easy to see that this property characterizes $\mathcal{M}$.


