Is Coherent sheaf acyclic?

Question

I am no expert in sheaf theory so the following question may be trivial. Let $X$ a complex manifold, and let $\mathcal{F}$ a coherent sheaf on $X$. Is $\mathcal{F}$ acyclic? If not: can you give a counterexample?

I don't know if somebody else asked the same question: in that case sorry for repetition.

Answer 1

The simplest counterexample is the sheaf of holomorphic differential $1$-forms $\Omega^1_{\mathbb P^1}=\mathcal O_{\mathbb P^1}(-2)$ on the Riemann sphere $\mathbb P^1=\mathbb P^1(\mathbb C)$.
It is coherent but not acyclic since $$H^1(\mathbb P^1,\Omega^1_{\mathbb P^1})= \mathbb C$$