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A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold.

The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the following reason: since $X$ is a Stein manifold, Cartan's Theorem B applies. Hence, its Picard group is isomorphic to its second cohomology group with integer coefficients $H^2(X,\mathbb{Z})$. The latter is non-trivial since $X$ has the homotopy of $S^1 \times S^1$.

I would like to see an explicit example of such a line bundle, as I was not able to produce one myself. I tried to write down some bundles arising from divisors, but I don't have a good intuition as to which divisors will give me non-trivial line bundles.

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    $\begingroup$ @abx I'm confused. How is that possible if $\Delta=\{x=y\}$ is a principal divisor? $\endgroup$
    – Piotr Achinger
    Commented Jan 24, 2022 at 11:45
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    $\begingroup$ Take a look at the first chapter of Mumford's "Abelian varieties", it will give you a hint how to classify line bundles on this torus using the universal cover and some group cohomology. $\endgroup$
    – Piotr Achinger
    Commented Jan 24, 2022 at 11:50
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    $\begingroup$ Note that every algebraic line bundle on $X$ (viewed as the torus $\mathbb{G}_{m,\mathbb{C}}^2$) is trivial since $\mathbb{C}[x^{\pm1},y^{\pm1}]$ is a UFD. $\endgroup$
    – Laurent Moret-Bailly
    Commented Jan 24, 2022 at 12:51
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    $\begingroup$ If you take a product of annuli instead, then a concrete example of such a divisor is described nicely in Section VI.5.2 ("Oka's counterexample") of Ranges book "Holomorphic Functions and Integral Representations in Several Complex Variables". $\endgroup$
    – Richard Lärkäng
    Commented Jan 24, 2022 at 13:00
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    $\begingroup$ @Piotr Achinger: Oops, you are right of course. I delete my comment. $\endgroup$
    – abx
    Commented Jan 24, 2022 at 13:23

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