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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, $\mathscr{P}(D)$

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up vote 2 down vote accepted

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?

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