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my problem is the following: i have a finite sequence of holomorphic $k-$forms $\alpha_{k}$, each defined on open subsets $U_{k} \subset M$, where $M$ is a complex $n$-dimensional manifold, such that $\bigcup_{k}U_{k} = M$. is there any method of gluing the holomorphic $k$-forms to give one global holomorphic $k-$form on $M$. Or if this is not possible, is there any possibility of gluing them that weakens the holomorphicity of the globaly defined $k$-form ? maybe meromorphic ? I hope I've presented my question clearly, and hope for a response. Many thanks in advance.


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Are you assuming that $\alpha_k$ and $\alpha_\ell$ agree on $U_k\cap U_\ell$? In this case, certainly yes. – Donu Arapura Aug 14 '11 at 19:46
And in the contrary case, certainly no. – Andreas Blass Aug 14 '11 at 19:51
Yes but if they do not agree on the cuts, is there any possibility to glue them together? – dimitry Aug 14 '11 at 19:51
I'm not sure what "glue together" would mean then. You'd have to modify them first, but I'm not sure what you allow. – Donu Arapura Aug 14 '11 at 20:12
Just to make an example, take $M=\mathbb{CP}^1=\mathbb{C}\cup\\{\infty\\}$. Take $U_1=\mathbb{C}$ and $U_2=\mathbb{C}^*\cup\\{\infty\\}$, $\alpha_1=e^{z}dz$, $\alpha_2=e^{1/z}dz$. How could you glue them? What conditions of "compatibility" would you like to ask for? I mean, what should the link between the data and the result be, apart from the latter being a holomorphic (or meromorphic) form? – Samuele Aug 14 '11 at 20:53

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