Appell–Humbert theorem

Template:Short description In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety. It was proved for 2-dimensional tori by Template:Harvs and Template:Harvs, and in general by Template:Harvs

Statement

Suppose that $T$ is a complex torus given by $V/\mathrm{\Lambda }$ where $\mathrm{\Lambda }$ is a lattice in a complex vector space $V$. If $H$ is a Hermitian form on $V$ whose imaginary part $E=\text{Im}(H)$ is integral on $\mathrm{\Lambda }\times \mathrm{\Lambda }$, and $\alpha$ is a map from $\mathrm{\Lambda }$ to the unit circle $U(1)=\{z\in ℂ:|z|=1\}$, called a semi-character, such that

$\alpha (u+v)=e^{i\pi E(u,v)}\alpha (u)\alpha (v)$

then

$\alpha (u)e^{\pi H(z,u)+H(u,u)\pi /2}$

is a 1-cocycle of $\mathrm{\Lambda }$ defining a line bundle on $T$. For the trivial Hermitian form, this just reduces to a character. Note that the space of character morphisms is isomorphic with a real torus

${\text{Hom}}_{\mathbf{\text{𝐀𝐛}}}(\mathrm{\Lambda },U(1))\cong {ℝ}^{2n}/{\mathbb{Z}}^{2n}$

if $\mathrm{\Lambda }\cong {\mathbb{Z}}^{2n}$ since any such character factors through $ℝ$ composed with the exponential map. That is, a character is a map of the form

$\text{exp}(2\pi i⟨l^{\ast },-⟩)$

for some covector $l^{\ast }\in V^{\ast }$. The periodicity of $\text{exp}(2\pi if(x))$ for a linear $f(x)$ gives the isomorphism of the character group with the real torus given above. In fact, this torus can be equipped with a complex structure, giving the dual complex torus.

Explicitly, a line bundle on $T=V/\mathrm{\Lambda }$ may be constructed by descent from a line bundle on $V$ (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms $u^{\ast }{𝒪}_V\to {𝒪}_V$, one for each $u\in \mathrm{\Lambda }$. Such isomorphisms may be presented as nonvanishing holomorphic functions on $V$, and for each $u$ the expression above is a corresponding holomorphic function.

The Appell–Humbert theorem Template:Harv says that every line bundle on $T$ can be constructed like this for a unique choice of $H$ and $\alpha$ satisfying the conditions above.

Ample line bundles

Lefschetz proved that the line bundle $L$, associated to the Hermitian form $H$ is ample if and only if $H$ is positive definite, and in this case $L^{\otimes 3}$ is very ample. A consequence is that the complex torus is algebraic if and only if there is a positive definite Hermitian form whose imaginary part is integral on $\mathrm{\Lambda }\times \mathrm{\Lambda }$

See also

• Complex torus for a treatment of the theorem with examples

References

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