Sheaf of closed holomorphic one-forms (${\mathcal{Z}}_M^1$)

The sheaf is ${\mathcal{Z}}_M^1=\ker (\partial :{\Omega }_M^1\to {\Omega }_M^2)$. The holomorphic Poincaré lemma gives $0\to \underline{\mathbb{C}}\to {\mathcal{O}}_M\stackrel{\partial }{\to }{\mathcal{Z}}_M^1\to 0$. On a compact complex surface its long exact sequence in sheaf cohomology yields $0\to H^0(M,{\Omega }_M^1)\to H^1(M,\mathbb{C})\to H^1(M,{\mathcal{O}}_M)$.