= Solution
Let $\mathcal Z_M^1$ be the <sheaf of closed holomorphic one-forms>, namely $\ker(\partial:\Omega_M^1\to\Omega_M^2)$. The local exactness proved in part (b) gives the <short exact sequence of sheaves>
$$
0\longrightarrow\underline{\mathbb C}\longrightarrow\mathcal O_M\xrightarrow{\partial}\mathcal Z_M^1\longrightarrow0.
$$
Its <long exact sequence in sheaf cohomology> begins
$$
H^0(M,\mathbb C)\longrightarrow H^0(M,\mathcal O_M)
\longrightarrow H^0(M,\mathcal Z_M^1)
\xrightarrow{\delta}H^1(M,\mathbb C)\longrightarrow H^1(M,\mathcal O_M).
$$
The first arrow is an isomorphism: on a compact <complex manifold>, a global <holomorphic function> is constant on each <connected component>, just as a global section of the <constant sheaf> is. Thus $\delta$ is injective. By part (c), every global <holomorphic differential form> of degree one is closed, so $H^0(M,\mathcal Z_M^1)=H^0(M,\Omega_M^1)$.
Concretely, for a <holomorphic differential form> of degree one choose local holomorphic primitives $f_i$. The connecting class $\delta(\alpha)$ is represented by the locally constant <Čech cocycle> $f_j-f_i$ on overlaps. The following arrow is induced by the inclusion of the <constant sheaf> into the <sheaf of holomorphic functions>. Exactness is inherited from the long exact sequence, giving
$$
\boxed{0\longrightarrow H^0(M,\Omega_M^1)\xrightarrow{\delta}H^1(M,\mathbb C)
\longrightarrow H^1(M,\mathcal O_M).}
$$
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