# Coherent locally free sheaves on projective varieties

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 $$H^0$$.

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.
• do you think this can be done for smooth proper schemes? The Picard group has to be non-trivial but I am not sure if you have large $H^0$ line bundles. – user138661 Apr 24 '19 at 11:59