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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]

The Appell–Humbert theorem identifies the positive Hermitian form $H_1$ with a holomorphic line bundle whose First Chern class is represented, in our linear-first convention, byIts integral over the square fundamental domain is $1$, so $\mathrm{deg}L=1$. In particular the oriented Chern pairing is $-E_1(1,i)=1$; using $E_1(1,i)$ without adjusting the Hermitian convention would give the wrong sign.

The complex torus has a nowhere-vanishing differential $dz$, so its canonical divisor is trivial and its genus is one. A nonzero section of a line bundle of negative degree would have an effective divisor of negative degree, which is impossible. Thus $H^0(X,L^{-1})=0$. The Riemann-Roch theorem and Serre duality giveTherefore $\boxed{h^0(X,L)=1}$. The choice of semicharacter of a complex lattice changes the degree-zero component of the line bundle, but not this calculation.

Pulling back the factors of automorphy in the Appell–Humbert theorem by $z\mapsto cz$, where $c=a+bi$, pulls back both the Hermitian form and the semicharacter of a complex lattice. Thus the explicit answer isWriting $\lambda =m+ni$ makes the second formula completely explicit:The second equality is an equality of signs, obtained by reducing the exponents modulo $2$. The Hermitian form follows from $H_1(cz,cw)=|c{|}^2{H}_1(z,w)$. Its integer parameter equals the norm of the Gaussian integer, agreeing with the degree of this nonzero isogeny of elliptic curves. The formulas also include $c=0$, when the pulled-back line bundle is trivial.

Let $X=V/\Gamma$ be a complex torus. A Riemann form on a complex torus is a Hermitian form $H$ on $V$ whose imaginary parttakes integer values on $\Gamma \times \Gamma$. Equivalently, $E$ is an integral alternating form satisfyingfor every nonzero $v$. A polarisation of a complex torus is such a positive Riemann form, or equivalently its integral cohomology class. By the Appell–Humbert theorem, it is the first Chern class of an ample holomorphic line bundle. A polarisation is principal when the homomorphism $\Gamma \to {\Gamma }^{\ast }$ induced by $E$ is an isomorphism.

For an integral alternating form $E$ arising as the imaginary part of a Hermitian form, a semicharacter satisfies $\alpha (\lambda +\mu )=e^{\pi iE(\lambda ,\mu )}\alpha (\lambda )\alpha (\mu )$. The Appell–Humbert theorem uses this datum together with the Hermitian form to specify a holomorphic line bundle on a complex torus. The ratio of two semicharacters is an ordinary unitary group homomorphism. On the square period lattice, with $H(z,w)=z\overline{w}$, all semicharacters are $\alpha (m+ni)=(-1{)}^{mn}{u}^m{v}^n$ for $u,v\in U(1)$.