Appell–Humbert theorem

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

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

then

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

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

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

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

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

for some covector ${\displaystyle l^{*}\in V^{*}}$. The periodicity of ${\displaystyle {\text{exp}}(2\pi if(x))}$ for a linear ${\displaystyle f(x)}$ 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 ${\displaystyle T=V/\Lambda }$ may be constructed by descent from a line bundle on ${\displaystyle V}$ (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms ${\displaystyle u^{*}{\mathcal {O}}_{V}\to {\mathcal {O}}_{V}}$, one for each ${\displaystyle u\in U}$. Such isomorphisms may be presented as nonvanishing holomorphic functions on ${\displaystyle V}$, and for each ${\displaystyle u}$ the expression above is a corresponding holomorphic function.

The Appell–Humbert theorem (Mumford 2008) says that every line bundle on ${\displaystyle T}$ can be constructed like this for a unique choice of ${\displaystyle H}$ and ${\displaystyle \alpha }$ satisfying the conditions above.

Lefschetz proved that the line bundle ${\displaystyle L}$, associated to the Hermitian form ${\displaystyle H}$ is ample if and only if ${\displaystyle H}$ is positive definite, and in this case ${\displaystyle L^{\otimes 3}}$ 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 ${\displaystyle \Lambda \times \Lambda }$

• Complex torus for a treatment of the theorem with examples

• 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, Providence, R.I.: American Mathematical Society, 22 (3): 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, Providence, R.I.: American Mathematical Society, 22 (4): 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