Question on Kahler geometry: Kahler form and $\mathcal{O}_X(1)$

Question

Given a projective surface (2 complex dimensions) $X$ we can equip it with the line bundle $\mathcal{O}_X(1)$. Let us fix this line bundle. Recall that all projective surfaces are Kahler surfaces. Then, I heard in a seminar that for the class of the associated Kahler form $J$ it is true that $$ [J] = c_1(\mathcal{O}_X(1)) $$ Why is this the case (if I formulated correctly)? Could you explain such a relation?

Answer 1

Because you're restricting the hyperplane class (which corresponds to the sheaf $\mathscr O_{\Bbb P^n}(1)$) in $\Bbb P^n$ to $X$ to get the the sheaf $\mathscr O_X(1)$, and $c_1$ of the hyperplane class bundle is precisely the Kähler form on $\Bbb P^n$. So it's just naturality, pulling back by the inclusion map in both cases.