Question

Let $X$ be a projective complex manifold. Is it true that any coherent sheaf $\mathcal{E}$ on $X$ whose Hilbert polynomial is a constant has a surjective morphsim $\mathcal{O}_X\rightarrow \mathcal{M}$? Or dually is it true that any coherent sheaf $\mathcal{F}$ whose Hilbert polynomial is different from that of $\mathcal{O}_X$ merely by a constant has a injective morphism $\mathcal{F}\rightarrow \mathcal{O}_X$?

Answer 1

No. For the first question, let $X$ be a point and $\mathcal M=\mathbb C^2$ (vector space) or any $X$ and $\mathcal M$ be a vector space of dimension $\ge 2$ supported only at $x$.

For the second question, consider a Riemann surface $X$ and $\mathcal F$ a line bundle. Then the Hilbert polynomial of $\mathcal F$ is that of $\mathcal O_X$ plus the degree of $\mathcal F$, but $\mathcal F$ can inject into $\mathcal O_X$ only if its degree is not positive.