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Appell–Humbert theorem

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

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

then

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

is a 1-cocycle of Λ {\displaystyle \Lambda } defining a line bundle on T {\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

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

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

exp ( 2 π i ⟨ l ∗ , − ⟩ ) {\displaystyle {\text{exp}}(2\pi i\langle l^{*},-\rangle )}

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

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

Ample line bundles

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

See also

References

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