I think that I do not understand something in the definition of the line bundles. Line bundles have fibers of rank 1, that is isomorphic to $\mathbb{C}$. But I do not know how to connect this vector space with elements like $dz_i$, $\,dz_{i_1}\wedge dz_{i_{2}},$ etc. Here, $dz_i$ are locally defined one-forms for the coordinates $\{z_i\}$ of some patch of the manifold of complex dimension $m$.
So, why is the $m$-th product of forms a line bundle? What is the correspondence with $\mathbb{C}$? Can you give an example?
Also, what kind of bundle is then the collection of elements $\{dz_i\}$, $\{dz_{i_1}\wedge dz_{i_2} \}$ and so on?