Appell–Humbert theorem

Short description: Describes the line bundles on a complex torus or complex abelian variety

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 $V/\mathrm{\Lambda }$ 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 $U(1)=\{z\in \mathbb{ℂ}:|z|=1\}$, 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

$\mathrm{\Lambda }$

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

${\text{Hom}}_{\text{<mrow data-mjx-texclass="ORD">Ab</mrow>}}(\mathrm{\Lambda },U(1))\cong {\mathbb{ℝ}}^{2n}/{\mathbb{\mathbb{Z}}}^{2n}$

if

$\mathrm{\Lambda }\cong {\mathbb{\mathbb{Z}}}^{2n}$

since any such character factors through

$\mathbb{ℝ}$

composed with the exponential map. That is, a character is a map of the form

$\text{exp}(2\pi i⟨l^{*},-⟩)$

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 $T=V/\mathrm{\Lambda }$ 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 $u^{*}{{𝒪}}_V\to {{𝒪}}_V$, 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 (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 $\mathrm{\Lambda }\times \mathrm{\Lambda }$

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, http://gallica.bnf.fr/ark:/12148/bpt6k107455z.image.f159.langFR 
• Humbert, G. (1893), "Théorie générale des surfaces hyperelliptiques", Journal de Mathématiques Pures et Appliquées, Série IV 9: 29–170, 361–475, http://gallica.bnf.fr/ark:/12148/bpt6k107457q.image.f29.langFR 
• Lefschetz, Solomon (1921), "On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 22 (3): 327–406, doi:10.2307/1988897, ISSN 0002-9947 
• Lefschetz, Solomon (1921), "On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 22 (4): 407–482, doi:10.2307/1988964, ISSN 0002-9947 
• Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, OCLC 138290