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Let $X$ be a compact complex manifold and $Y$ a complex sub manifold of codimension $\ge 2$. If $\pi : X_{Y} \mapsto X$ is the blow-up of $X$ along $Y$, do you have any references for this result :

-$Pic(X_{Y}) = \pi^{*}Pic(X) \otimes \mathbb{Z}$? (whithout using "pure" algebraic geometry arguments but only complex geometry).

Here $Pic(X) = H^{1}(X, \mathcal{O}_{X}^{*})$ are all the classes of isomorphism of holomorphic line bundles over $X$.

I thank you in advance and wish you a good day.

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  • $\begingroup$ You don’t mean $\otimes$ :) $\endgroup$ Jul 29, 2022 at 17:27
  • $\begingroup$ Yes, thank you very much :). $\endgroup$
    – Analyse300
    Jul 29, 2022 at 17:48
  • $\begingroup$ The answer can be found here math.stackexchange.com/questions/4349970/… $\endgroup$
    – sti9111
    Jul 29, 2022 at 18:07
  • $\begingroup$ It uses algebraic geometry arguments but the parallel is clear for me. Thank you very much. $\endgroup$
    – Analyse300
    Jul 30, 2022 at 9:36

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