Hello everyone,
it seems to be "well-known" that $H^0(X;End(V))$ only contains isomorphisms where $X$ is a Riemann surface and $V$ a stable (algebraic) vector bundle over $X$. The usual proof considers (roughly) the (coherent) image sheaf of a non-zero vector bundle morphism $\varphi:V\to V$ and one obtains a contradiction under the hypothesis that $im(\varphi)\neq V$ : $\mu(V)<\mu(im(\varphi))<\mu(V)$ (where $\mu(V)=deg(V)/rk(V)$ denotes the slope of $V$).
Now my question is if this assertion is still true in the complex (differential) geometric context, i.e. when one defines holomorphicity (and hence stability) via del-bar-operators.
The previous proof doesn't seem to work since the image of a (smooth) vector bundle homomorphism $\varphi:V\to V$ is in general NOT a vector (sub)bundle of $V$ (unless the rank of $\varphi$ is (locally) constant). So $\mu(im(\varphi))$ doesn't make sense since the image of $\varphi$ is in general not a subbundle of $V$.
Or am I missing something in the holomorphic setup?
Since I'm not an expert in algebraic geometry I have some difficulties in "translating" results concering stable (algebraic) vector bundles over a Riemann surface into the complex geometric (del-bar) approach to stable (holomorphic) vector bundles.
Thanks in advance for any answers, comments and remarks!
Edit: I changed my terminology from "differential geometric (dg)" to "complex geometric" which seems more appropriate (thanks @David Roberts!) and added some comments. Hopefully my questions/confusion is more understandable now.
Best Answer
Consider the two following potential definitions of stability of a locally free coherent sheaf $E$.
(A) Every subbundle (i.e. locally free subsheaf) has strictly smaller slope.
(B) Every subsheaf has strictly smaller slope.
I claim that over a curve $X$ these two definitions are equivalent. Indeed, suppose (A) holds, and let $F\subset E$ be a subsheaf. Consider the sequence
$$0\to F\to E \to Q\to 0.$$
Here $Q$ may fail to be locally free; Put $G = Q/Q_{tors}$, which is then locally free since $X$ is a curve. Define $K$ by the sequence
$$0\to K \to E \to G\to 0,$$
and observe that $K$ is locally free of the same rank as $F$. Furthermore, $c_1(K) \geq c_1(F)$ since $c_1(Q_{tors})\geq 0$, so $\mu(F) \leq \mu(E)$.
Thus it is equivalent to only consider subbundles for stability, so long as we are working on a curve. Together with GAGA, this shows the two approaches (and any possible combination of definitions) are essentially the same.