Abelian Varieties

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 25 June 2012

The moduli space of g-dimensional complex tori

A g-dimensional complex torus is a g-dimensional compact complex manifold g/Λ where Λ = -span { a1,a2,...,a2g } with a1,...,a2gg. The period matrix of g/Λ is Ω = - a1 - - a2 - - a2g - M2g×g(). The matrix Ω is rank 2g (i.e. Λ has rank 2g) if and only if det(Ω,Ω_) 0. Let = { ΩM2g×g() | det(Ω,Ω_)0 }. Then GL2g \/GLg() { isomorphism classes of g-dimensional complex tori } Ω g/Λ, where Λ = -span(Ω). Each double coset in GL2g() \/GLg() has a representative of the form Ω = τIg with τMg×g() and det(Imτ) 0. The action of GL2g () on coset representatives is Mτ = (Aτ+B) (Cτ+D)-1 for M = A B C D GL2g().

Notes and References

This section follows [SU, §2.1.1].

The moduli space of polarized abelian varieties

An abelian variety is a complex torus A such that there exists an embedding A&hkrarr;d().

A polarized abelian variety is a pair (A,) where A is a complex torus and is an ample line bundle on A.

In this case A(V) where A = H0(A,).

(Appell-Humbert theorem). Let A = g/Λ be a g-dimensional complex torus and let be a line bundle on g/Λ.

  1. = g ×Λ = g× (z,c) = ( z+μ, χ(μ) e π( z,μ + 12 μ,μ ) c ) for μΛ for a unique Hermitian form A,A: g×g and function χ:ΛU1() satisfying
    1. if μ,νΛ then
    2. Im(μ,ν) and χ(μ+ν) = χ(μ) χ(ν) e iπIm( μ,ν ) .
  2. The line bundle is ample if and only if A,A: g×g is positive definite.

The Siegel upper half space is 𝒢g = { τMg×g() | τt=τ, Imτ>0 }. If A = g/Λ and is an ample line bundle on A then there exists a basis γ1,...,γ2gΛ and unique d1,...,dg >0 with d1|d2||dg so that Λ = -span {a1,...,a2g} and z,w = z(Imτ)-1 w_t (AbV 1) where, in the basis 1d1γg+1,   1d2γg+2,   ...,   1dgγ2g of g Ω = - a1 - - a2 - - a2g - = τΔ with   τ𝒢g   and   Δ = d1 0 0 dg . The freedom in the choice of the basis γ1,...,γ2g is controlled by the paramodular group Sp(Δ,) = { MGL2g() | M 0 Δ -Δ 0 Mt = 0 Δ -Δ 0 }. Define an action of Sp(Δ,) on 𝒢g by Mτ = ( ατ+βΔ ) ( γτ+δΔ )-1 Δ, if M = α β γ δ Sp(Δ,). (AbV 2)

Let d1,...,dg >0 with d1|d2||dg. For τ𝒢g let Λ and be determined from τ by (AbV 1) and part (a) of the Appell-Humbert theorem with χ: ΛU1() given by χ(μ) = ?????

  1. The map Sp(Δ,) \ 𝒢g { isomorphism classes of abelian varieties with Δ-polarization } τ ( g/Λ, ) is a bijection.
  2. dim( H0( g/Λ, ) ) = d1d2dg.

Notes and References

The Hermitian form in the Appell-Humbert theorem in the "first Chern class of ", see [SU, (2.15)].

The Imτ>0 condition in 𝒢g is what is providing the positive definiteness of A,A (and hence the ampleness of ).

The action of Sp(Δ,) on 𝒢g given in (AbV 2) is a generalization of the action of Sp2() = SL2() on the upper half plane by Möbius transformations.

The integral symplectic group of degree g, or Siegel modular group of degree g is Sp2g() = Sp(Id,).

This section follows [SU, §2.1].

These notes were typed from handwritten notes by Arun Ram written in May/June 2012.


[SU] Y. Shimizu and K. Ueno, Advances in Moduli Theory, Translations of Mathematical Monographs Vol. 206, American Mathematical Soc., 2002. ISSN 0065-9282 v.206.

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