Appell–Humbert theorem

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 and , and in general by

Statement

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

:$\alpha(u+v) = e^\alpha(u)\alpha(v)\$

then

:$\alpha(u)e^\$

is a 1- cocycle of $\Lambda$ 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

$\text_(\Lambda,U(1)) \cong \mathbb^/\mathbb^$

if $\Lambda \cong \mathbb^$ since any such character factors through $\mathbb$ composed with the exponential map. That is, a character is a map of the form

$\text(2\pi i \langle l^*, -\rangle )$

for some covector $l^* \in V^*$. The periodicity of $\text(2\pi i f(x))$ 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 $T = V/\Lambda$ 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 $u^*\mathcal_V \to \mathcal_V$, one for each $u \in U$. 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 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 $L^$ 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 $\Lambda\times\Lambda$

See also

* Complex torus for a treatment of the theorem with examples

References

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{{DEFAULTSORT:Appell-Humbert theorem
Abelian varieties
Theorems in algebraic geometry
Theorems in complex geometry