Let $X$ be a nonsingular projective variety with dimension $n$, $Z$ is a closed subvariety of $X$ with dimension $n-1$. there is a morphims $f:H^{n-1}(X,\Omega_{X}^{n-1})\longrightarrow$$H^{n-1}(Z,\Omega_{Z}^{n-1})$, by the Serre's duality, there is a cohomology class $\zeta(Z) \in$$H^{1}(X,\Omega_{X}^{1})$.

On the other hand,$dlog$:$O_{X}^{*}\longrightarrow$ $\Omega_{X}^{1}$, then we have $c :H^{1}(X,O_{X}^{*})\longrightarrow$$H^{1}(X,\Omega_{X}^{1})$.

My questions are:

Why the $c(O_{X}(Z))=\zeta(Z)$?

How to understand the morphism $dlog$ and $c$?

Thank you very much.