Consider the Riemann sphere $S^2$, together with the sheaf $\Omega_{S^2}$ which is defined to be $\Omega_{S^2}:=\operatorname{Ker}\overline{\partial}$ with $\overline{\partial}\colon\mathcal A^{1,0}\to \mathcal A^2$ and I want to show $H^0(S^2,\Omega_{S^2})=0$ without using the exact sequence $$0\to \mathbb C\to \mathcal O_{S^2}\to \Omega_{S^2}\to 0$$ or Hodge symmetry.
I have another tool which is the exact sequence $$0\to \Omega_{S^2}\to \mathcal A^{1,0}\to \mathcal A^2\to 0$$ where $\mathcal A^2$ is the sheaf of complex 2-forms ($\mathcal A^2=\mathbb C\otimes_{\mathbb R}\mathcal F^2_{\mathbb R})$, $\mathcal F^2_{\mathbb R}$ the sheaf of real 2-forms) and $\mathcal A^{1,0}$ is defined to be the kernel of $i\times id-I\colon \mathcal A^1\to \mathcal A^1$, where $I$ is a $\mathcal A^0$-linear endomorphism such that it commutes with conjugation and $I(df)=idf$ for $f\colon U\to \mathbb C$ holomorphic. Here I think the right map is the standard differential $d$. From this we can take the long exact sequence but I have no idea what the cohomology of the sheaves of differentials are.
Another idea might be to find an acyclic resolution of $\Omega_{S^2}$ and compute cohomology directly. If the above sequence is an acyclic resolution we want to show that $d\colon \Gamma(S^2,\mathcal A^{1,0})\to \Gamma(S^2,\mathcal A^2)$ is injective , but $d(fdz)=0\iff df\wedge dz=0\iff \frac{\partial f}{\partial \overline{z}}d\overline{z}\wedge dz=0$ but then $\frac{\partial f}{\partial \overline{z}}=0$ which isn't enough it is ? And is the sequence an acyclic resolution ?
I'm a bit lost with this and I didn't find it in books about the subjects which often use different definitions. Any help on this ?