Exercise 4.2 in Voisin's "Hodge Theory ... I" refers to
the class in $H^1(X,K_X)$ of the form $\bar \partial \mu_i$, where $\mu_i$ is a differential form of type $(1,0)$, which is $C^{\infty}$ away from $x_i$, and equal to $\frac{dz_i}{z_i}$ in a neighborhood of $x_i$.
Here, $X$ is a compact complex curve, $K_X$ is the sheaf of holomorphic $1$-forms, $x_i$ is a point on $X$, and $z_i$ is a local coordinate centered at $x_i$.
How does $\mu_i$ define such a cohomology class?