Let $k$ be a field, $X$ be a connected smooth projective $k$-scheme. Let $f:X\rightarrow X$ be a finite $k$-morphism that is surjective on the underlying topological spaces. Suppose $f$ has degree $n$. 


Is it possible that there does not exist a coherent locally free sheaf $F$ on $X$ such that $H^0(X, F)=n\,\mathrm{rank}(F)$? 


I think for curves, one can find line bundles of any $H^0$.



Coherent locally free sheaves on projective varieties
You can always find such a sheaf.


Take a very ample line bundle $L$ on $X$, with $H^0(X, \, L)=a \geq n$, and set $$F:=L^{\oplus n-1} \oplus \mathcal{O}\_X^{\oplus a-n}.$$ 


Then $$H^0(X,\, F)=n(a-1)=n \; \mathrm{rank}(F).$$
Furthermore, the role of $f \colon X \to X$ is irrelevant here.


**Remark.** It seems to me that an interesting question is the existence of an *irreducible* locally free sheaf on $X$ with the same property. 


