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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:The corresponding bundle $L(H,\alpha )$ is the quotient of $V\times \mathbb{C}$ by the group actionThe 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.^[ref]

Tensor products add the Hermitian forms and multiply the semicharacters: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.

Christian Schnell studies Hodge theory and the geometry and topology of complex algebraic varieties. His Stony Brook research page emphasizes applications of mixed Hodge modules to algebraic geometry.^[ref]

His lecture on the Appell–Humbert theorem describes holomorphic line bundles on a complex torus using a Hermitian form and compatible lattice data. Tensor product becomes addition of the forms and multiplication of the accompanying characters. The classification turns an abstract Picard group into explicit data that also help decide when a torus is projective.^[ref]