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I am currently reading a paper by Mustata and Popa on the Van de Ven theorem, you can read the paper here.

On page 51 there is the following map $$\alpha_\mathbb{C} : \mathrm{Num}(X) \otimes_\mathbb{Z} \mathbb{C} \hookrightarrow H^1(X,\Omega^1_X)$$ where $X$ is a projective variety, $\mathrm{Num}(X)$ is the group of divisors on $X$ up to numerical equivalence and $\Omega^1_X$ its cotangent bundle.

I would like to understand where this map comes from. Here is what I already know:

  • $\mathrm{Num}(X)$ is a quotient of $\mathrm{Pic}(X)$
  • $\mathrm{Pic}(X) \cong H^1(X,\mathcal{O}^*_X)$
  • The map \begin{align*}\mathrm{d}(\log):\mathcal{O}^*_X&\longrightarrow\Omega^1_X \\ f&\longmapsto f^{-1}\mathrm{d}f\end{align*} induces a well defined map $c : H^1(X,\mathcal{O}^*_X) \to H^1(X,\Omega^1_X).$
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    $\begingroup$ The map $c$ associates to a line bundle its first Chern class. It is well-known (and easy) that it factors through numerical equivalence. $\endgroup$
    – abx
    Commented Aug 30 at 15:28
  • $\begingroup$ @abx Do you have a reference for that? $\endgroup$
    – Adil Raza
    Commented Aug 30 at 15:59

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