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 {{harvs|txt|last=Appell|authorlink=Paul Émile Appell|year=1891}} and {{harvs|txt|last=Humbert|authorlink=Marie Georges Humbert|year=1893}}, and in general by {{harvs|txt|last=Lefschetz|authorlink=Solomon Lefschetz|year=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

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

\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 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 $u^*\mathcal{O}_V \to \mathcal{O}_V$ , 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 {{harv|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 $\Lambda\times\Lambda$

References

{{citation|authorlink=Paul Émile Appell|first=P.|last=Appell|title=Sur les functiones périodiques de deux variables|journal=Journal de Mathématiques Pures et Appliquées |series=Série IV|volume=7|pages=157–219|year=1891|url=http://gallica.bnf.fr/ark:/12148/bpt6k107455z.image.f159.langFR}}
{{citation|first=G.|last=Humbert|title=Théorie générale des surfaces hyperelliptiques|journal=Journal de Mathématiques Pures et Appliquées |series=Série IV|volume=9|pages=29–170, 361–475|year=1893|url=http://gallica.bnf.fr/ark:/12148/bpt6k107457q.image.f29.langFR}}
{{Citation | last1=Lefschetz | first1=Solomon | title=On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties | jstor=1988897 | publisher=American Mathematical Society | location=Providence, R.I. | year=1921 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=22 | issue=3 | pages=327–406 | doi=10.2307/1988897| doi-access=free }}
{{Citation | last1=Lefschetz | first1=Solomon | title=On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties | jstor=1988964 | publisher=American Mathematical Society | location=Providence, R.I. | year=1921 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=22 | issue=4 | pages= 407–482 | doi=10.2307/1988964}}
{{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Abelian varieties | origyear=1970 | publisher=American Mathematical Society | location=Providence, R.I. | series=Tata Institute of Fundamental Research Studies in Mathematics | isbn=978-81-85931-86-9 | oclc=138290 | mr=0282985 | year=2008 | volume=5}}

Category:Abelian varieties

Category:Theorems in algebraic geometry

Category:Theorems in complex geometry

Appell–Humbert theorem

Statement

Ample line bundles

See also

References