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Let $\mathbb C$ be a standard complex plane, $\Lambda=\mathbb Z+i\mathbb Z$ be a lattice in $\mathbb C$, then we can get a torus $T=\mathbb C/\Lambda$, when $T$ is equipped with a standard Kähler metric $h=dz\otimes d\bar z$, then its associated Kähler form is $\omega=\frac{i}{2}dz\wedge d\bar z$. According to Chern's book complex manifolds p.65, Formula 7.30, the Kähler class $[\omega]$ is integral (then $\omega$ is called a Hodge metric), i.e. $[\omega]\in H^{1,1}(X,\mathbb Z)$. Then using Lefschetz theorem on (1,1) class, we get there is a holomorphic line bundle $L$ over $T$ such that its Chern class is exactly $[\omega]$. But I find it hard to construct $L$ explictly, since we should give $T$ a covering and the transition functions between them, is there a canonical choice of covering?

Tom
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  • one relevant reference is Herbert Lange's Abelian Varieties over the Complex Numbers, section 1.2. – Invariance Jun 08 '23 at 16:32
  • @Invariance, thanks for pointing out this reference to me, in section 1.2.2, the author seems do not choose a covering, but uses a factor of automorphy. – Tom Jun 09 '23 at 03:59

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