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.

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This is Exercise III.7.4(c) in Hartshorne. – Sándor Kovács Jan 30 '12 at 9:55
I know this is Hartshorne's exercise, but I don't know how to prove it. – kiseki Jan 30 '12 at 11:13
Yet, this is not the right place to find out: see mathoverflow.net/faq#whatnot Perhaps you could ask your AG professor or a fellow graduate student. – Sándor Kovács Jan 30 '12 at 17:06
...or at math.stackexchange.com – Sándor Kovács Jan 30 '12 at 17:08
Sándor: From my experience on math.stackexchange.com, I would say that this question is likely too high-level to get a good answer there. (No one will think it is inappropriate, but no one will have a shot at answering it except people who are also on mathoverflow.) – Charles Staats Jan 30 '12 at 21:32