Sheaf of closed holomorphic one-forms
= Sheaf of closed holomorphic one-forms
{title2=$\mathcal Z_M^1$}
The sheaf is $\mathcal Z_M^1=\ker(\partial:\Omega_M^1\to\Omega_M^2)$. The <holomorphic Poincaré lemma> gives $0\to\underline{\mathbb C}\to\mathcal O_M\xrightarrow{\partial}\mathcal Z_M^1\to0$. On a compact complex surface its <long exact sequence in sheaf cohomology> yields $0\to H^0(M,\Omega_M^1)\to H^1(M,\mathbb C)\to H^1(M,\mathcal O_M)$.