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I'm reading the book "R. O. Wells Jr. - Differential Analysis on Complex Manifolds" and on page 247 the author claims two things about the stability of holomorphic vector bundles that I'm struggling to prove:

Here I'm talking about holomorphic vector bundles over a Riemann surface.

1)Every polystable bundle is in particular semistable.

2)If $F\subset E$ is a subbundle s.t $\mu(F)=\mu(E)$, then $E/F$ is semistable with $\mu(E)=\mu(E/F)$.

I think it should be easy to see, but as I'm not very familiar with this subject, I don't know where to start. Does anyone have any references or suggestions?

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For Question 1, let's take $E=E_1\oplus E_2$, and let $\mu$ be the slope of $E$ (which agrees with the slope of $E_1$ and $E_2$). Let $F\subset E$ be a subbundle and let $p:=\mathrm{pr_1}|_F:F\to E_1$ be the projection on the first factor. There is an exact sequence $$0\to \ker(p)\to F \to \mathrm{Im}(p) \to 0$$ Note that $\ker(p) \subset E_2$ and $\mathrm{Im}(p) \subset E_1$. By semistability of $E_1,E_2$ and the fact that the slope of the middle term of an exact sequence if a convex combination of the slopes of the other two terms, one finds $\mu(F)\le \mu$.

About question 2, this is not exactly what Wells is saying even though this is not relevant for your question. Take a semistable bundle E that is not stable. Then, take $F\subset E$ the subbundle with smallest rank among the subbundles that have same slope as $E$. Then $F$ is automatically stable. Moreover, as the slope of $E$ is a convex combination of the slope of $F$ and $E/F$, those three quantities must coincide. Finally, $E/F$ is semistable. To see that, use the characterization of semistability using quotient bundles instead of subbundles. Any quotient of $E/F$ is again a quotient of $E$. As $E$ and $E/F$ have the same slope, you are done.

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