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 Appell (1891) and Humbert (1893), and in general by Lefschetz (1921)

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 $E={\text{Im}}(H)$ is integral on $\Lambda \times \Lambda$, and $\alpha$ is a map from $\Lambda$ to the unit circle $U(1)=\{z\in \mathbb {C} :|z|=1\}$, 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 $\Lambda$ 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

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

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

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

for some covector $l^{*}\in V^{*}$. 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 $T=V/\Lambda$ 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 $u^{*}{\mathcal {O}}_{V}\to {\mathcal {O}}_{V}$, one for each $u\in \Lambda$. 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 (Mumford 2008) 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 $\Lambda \times \Lambda$

See also

• Complex torus for a treatment of the theorem with examples

References

• Appell, P. (1891), "Sur les functiones périodiques de deux variables", Journal de Mathématiques Pures et Appliquées, Série IV, 7: 157–219
• Humbert, G. (1893), "Théorie générale des surfaces hyperelliptiques", Journal de Mathématiques Pures et Appliquées, Série IV, 9: 29–170, 361–475
• Lefschetz, Solomon (1921), "On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties", Transactions of the American Mathematical Society, 22 (3), Providence, R.I.: American Mathematical Society: 327–406, doi:10.2307/1988897, ISSN 0002-9947, JSTOR 1988897
• Lefschetz, Solomon (1921), "On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties", Transactions of the American Mathematical Society, 22 (4), Providence, R.I.: American Mathematical Society: 407–482, doi:10.2307/1988964, ISSN 0002-9947, JSTOR 1988964
• Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290