= Appell–Humbert theorem
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For a <complex torus> $X=V/\Lambda$, every <holomorphic line bundle> is classified by a pair $(H,\alpha)$. Here $H$ is a <Hermitian form>, linear in its first argument, whose imaginary part $E$ is integral on $\Lambda\times\Lambda$, and $\alpha$ is a <semicharacter of a complex lattice>:
$$
\alpha(\lambda+\mu)=\alpha(\lambda)\alpha(\mu)e^{\pi iE(\lambda,\mu)}.
$$
The corresponding bundle $L(H,\alpha)$ is the quotient of $V\times\mathbb C$ by the <group action>
$$
\lambda\cdot(z,v)=
\left(z+\lambda,
\alpha(\lambda)e^{\pi H(z,\lambda)+\pi H(\lambda,\lambda)/2}v\right).
$$
The semicharacter identity makes successive lattice translations compose correctly. The resulting correspondence is an <isomorphism> onto the <Picard group>, so it gives both existence and uniqueness of the classification.<Lecture 4: The Appel-Humbert theorem by Christian Schnell>[Theorem 4.1, p.1]{ref}
<Tensor products> add the <Hermitian forms> and multiply the semicharacters:
$$
L(H_1,\alpha_1)\otimes L(H_2,\alpha_2)
\cong L(H_1+H_2,\alpha_1\alpha_2).
$$
When $H=0$, the semicharacter is an ordinary unitary <group homomorphism>, describing bundles with zero <First Chern class>. A positive-definite $H$ instead gives a <polarization of a complex torus>. The theorem therefore separates the discrete integral curvature data from the remaining flat holomorphic choices.
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