About first Chern class and Poincaré duality in case of an ample divisor

Question

Let $D$ be a very ample divisor in $X$ projective variety. I can't understand why the first chern class $c_1(\mathscr{O}_X(D))$ equals the Poincarè dual of D, $\mathscr{P}(D)$

Answer 1

Over the complex numbers this is the content of Proposition 4.4.13 in Huybrecht's book "Complex Geometry".

There this is proven for any divisor $D$ in any compact complex manifold $X$.

Were you explicitely looking for the general result?