Let $X$ be a smooth projective variety, say it is a K3 surface. Fix a Chern character $(ch_0,ch_1,ch_2)$. Then if we consider the global sections of all the possible (semi)stable vector bundles and unstable vector bundles. Then is it true that the number of possible global sections of (semi)stable vector bundles is smaller or equal to the possible global sections of unstable vector bundles with same character? My thought was considering a unstable vector bundle $F$, with Harder-Narashman factors $F_i$, then the global section of $F$ is smaller or equal than the sum of the global sections of the factors $F_i$, and equality can be achieved to consider direct sum of the factors. Then I want to deduce that a (semi)stable vector bundle $E$ with same Chern character of $F$, i.e., sum of Chern character of $F_i$, has smaller number of global sections than the sum of all $F_i$'s.
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$\begingroup$ Just a remark. Note that by Riemann-Roch, the Chern character of a vector bundle $V$ determines its Euler char. $\chi(V)=h^0(V)-h^1(V)+h^2(V)=h^0(V)-h^1(V)+h^0(V^\vee)$ (Serre duality). If $V$ is semistable of slope $>0$ then $h^0(V^\vee)=0$. So if $h^1(V)$ is fixed and the slope of $V$ is positive, the answer to your question is yes. $\endgroup$– Damian RösslerAug 19, 2021 at 10:06
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