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  • $\begingroup$ Can you give a reference for "Since $H^1(X,\mathcal{C}_X^*)$ is in bijection eith the set of isomorphism classes of complex line bundles over $X$" $\endgroup$ Commented Sep 8, 2018 at 17:50
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    $\begingroup$ I don't know a reference for this specific fact off the top of my head, but it follows easily from the cocycle description of line bundles. $\endgroup$ Commented Sep 8, 2018 at 18:27
  • $\begingroup$ :D Yes, that is true... I should not even asked for reference... Given a Line bundle $E\rightarrow X$ we have an open cover ${U_\alpha}$ for $X$ and functions $g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow \mathbb{C}^*$ that determines the line bundle.. This collection ${g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow \mathbb{C}^*}$ gives an element in first sheaf cohomology of $X$ with sheaf $\mathcal{C}_X^*$.. Similarly, given an element in $H^1(X,\mathcal{C}_X^*)$ it gives a collection of such transition functions, which will anyways determine a Line bundle over $X$... $\endgroup$ Commented Sep 9, 2018 at 1:46
  • $\begingroup$ Thus, there is a bijection between $H^1(X,\mathcal{C}_X^*)$ and the isomorphism classes of complex line bundles... Even I can not think of any reference where this is mentioned... $\endgroup$ Commented Sep 9, 2018 at 1:51
  • $\begingroup$ This will force the question how did some one guess there could be a bijection between $H^2(X,\mathbb{Z})$ and the collection of homotopy classes $X\rightarrow K(\mathbb{Z},2)$?... I see going from vector bundle to $H^2(X,\mathbb{Z})$ as natural but could not guess how one can see possibility of bijection from $H^2(X,\mathbb{Z})$ to $[X,\mathbb{C}\mathbb{P}^\infty]$... $\endgroup$ Commented Sep 9, 2018 at 2:33