Stability of complex vector bundles In the definition of the stability for a holomorphic vector bundle, where exactly do we use the fact that the bundle is holomorphic and not only complex? Why couldn't we define the slope of any complex vector bundle?
The only place I can see we actualy use holomorphicity is that we require that the slope should be smaller for any holomorphic subbundles. Is there anything that I am missing?
 A: I will try to give you an explanation only considering complex and holomorphic vector bundles over Riemann Surfaces. In general it is true the following:
Theorem
Let $X$ be a compact connected Riemann Surface of genus $g\ge 2$ and let $E,F$ be two smooth complex vector bundles over $X$. Denote with $r=$rk$(E)$, $ \ r'=$rk$(F)$, $ \ d=$deg$(E):=\int_Xc_1(E)$, $ \ d'=$deg$(F):=\int_Xc_1(F)$. Then $E$ and $F$ are isomorphic if and only if $r=r'$ and $d=d'$. Moreover for any pair $(r,d)\in\mathbb{Z}_+\times\mathbb{Z}$ there exist a smooth complex vector bundle over $X$ with rank $r$ and degree $d$
This classification can be found in:
[M. Thaddeus : An introduction to the topology of the moduli space
of stable bundles on a Riemann Surface, 1997]
In particular it says that from a smooth (topological) viewpoint, isomorphism classes of complex vector bundles over $X$ are parametrized by discret invariants. It must be noted that for this classification we didn't need to invoke the slope-stability.
Everything change if we now consider smooth complex vector bundle $E$ with an holomorphic structure, which can be given by an operator (this is not the usual definition but an equivalent one):
\begin{equation*}\bar{\partial}_E:\Omega^0(X;E)\longrightarrow\Omega^0(X;E\otimes (T^{0,1}X)^*)\end{equation*} such that:

*

*$\bar{\partial}_E\circ\bar{\partial}_E=0$

*$\bar{\partial}_E(f\cdot s)=\bar{\partial}f\cdot s + f\cdot\bar{\partial}_E(s), \qquad$ for $f\in C^\infty(X)$ and $s\in\Omega(X;E)$
If you try to classify all the possible holomorphic structure on $E$ up to isomorphism (two holomorphic structures are isomorphic if they are conjugated by an automorphism of $E$) then you can realize that strange things can happen. In fact there are examples in which the moduli space of holomorphic structures is not Hausdorff and hence it can not carry a manifold structure, which is exactly what one wants when defining a moduli space.
In order to solve this problem, the notion of slope-stability was introduced. In fact, thanks to the works of Mumford, Donaldson, Atiyah, Uhlenbeck and many other very good mathematicians fantastic results have been obtained regarding the moduli space of stable holomorphic bundles over projective manifolds.
