Abelian Varieties

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 ${ℂ}^{g}/\Lambda$ where $Λ = ℤ-span { a1,a2,...,a2g } with a1,...,a2g∈ℂg.$ The period matrix of ${ℂ}^{g}/\Lambda$ is $Ω = - a1 - - a2 - ⋮ - a2g - ∈ M2g×g(ℂ).$ The matrix $\Omega$ is rank $2g$ (i.e. $\Lambda$ has rank $2g$) if and only if $\mathrm{det}\left(\Omega ,\stackrel{_}{\Omega }\right)\ne 0.$ Let $ℳ = { Ω∈M2g×g(ℂ) | det(Ω,Ω_)≠0 }.$ Then $GL2g \ℳ/GLg(ℂ) →∼ { isomorphism classes of g-dimensional complex tori } Ω ↦ ℂg/Λ, where Λ = ℤ-span(Ω).$ Each double coset in ${\mathrm{GL}}_{2g}\left(ℤ\right)\ℳ/{\mathrm{GL}}_{g}\left(ℂ\right)$ has a representative of the form $Ω = τIg with τ∈Mg×g(ℂ) and det(Imτ) ≠ 0.$ The action of ${\mathrm{GL}}_{2g}\left(ℤ\right)$ on coset representatives is $M⋅τ = (Aτ+B) (Cτ+D)-1 for M = A B C D ∈ GL2g(ℤ).$

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 $A&hkrarr;{ℙ}^{d}\left(ℂ\right).$

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

In this case $A↪ℙ(V) where A = H0(A,ℒ).$

(Appell-Humbert theorem). Let $A={ℂ}^{g}/\Lambda$ be a $g-$dimensional complex torus and let $ℒ$ be a line bundle on ${ℂ}^{g}/\Lambda .$

1. $ℒ = ℂg ×Λ ℂ = ℂg×ℂ ⟨ (z,c) = ( z+μ, χ(μ) e π( ⟨z,μ⟩ + 12 ⟨μ,μ⟩ ) c ) for μ∈Λ ⟩$ for a unique Hermitian form $⟨A,A⟩: ℂg×ℂg → ℂ and function χ:Λ→U1(ℂ)$ satisfying
1. if $\mu ,\nu \in \Lambda$ then
2. $\mathrm{Im}\left(⟨\mu ,\nu ⟩\right)\in ℤ$ and $\chi \left(\mu +\nu \right)=\chi \left(\mu \right)\chi \left(\nu \right){e}^{i\pi \mathrm{Im}\left(⟨\mu ,\nu ⟩\right)}.$
2. The line bundle $ℒ$ is ample if and only if $⟨A,A⟩: ℂg×ℂg → ℂ$ is positive definite.

The Siegel upper half space is $𝒢g = { τ∈Mg×g(ℂ) | τt=τ, Imτ>0 }.$ If $A={ℂ}^{g}/\Lambda$ and $ℒ$ is an ample line bundle on $A$ then there exists a basis ${\gamma }_{1},...,{\gamma }_{2g}\in \Lambda$ and unique ${d}_{1},...,{d}_{g}\in {ℤ}_{>0}$ with ${d}_{1}|{d}_{2}|\cdots |{d}_{g}$ so that $Λ = ℤ-span {a1,...,a2g} and ⟨z,w⟩ = z(Imτ)-1 w_t (AbV 1)$ where, in the basis of ${ℂ}^{g}$ The freedom in the choice of the basis ${\gamma }_{1},...,{\gamma }_{2g}$ is controlled by the paramodular group $Sp(Δ,ℤ) = { M∈GL2g(ℤ) | M 0 Δ -Δ 0 Mt = 0 Δ -Δ 0 }.$ Define an action of $\mathrm{Sp}\left(\Delta ,ℤ\right)$ on ${𝒢}_{g}$ by $M⋅τ = ( ατ+βΔ ) ( γτ+δΔ )-1 Δ, if M = α β γ δ ∈ Sp(Δ,ℤ). (AbV 2)$

Let ${d}_{1},...,{d}_{g}\in {ℤ}_{>0}$ with ${d}_{1}|{d}_{2}|\cdots |{d}_{g}.$ For $\tau \in {𝒢}_{g}$ let $\Lambda$ and $ℒ$ be determined from $\tau$ by (AbV 1) and part (a) of the Appell-Humbert theorem with $\chi :\Lambda \to {U}_{1}\left(ℂ\right)$ given by $\chi \left(\mu \right)=?????$

1. The map $Sp(Δ,ℤ) \ 𝒢g → { isomorphism classes of abelian varieties with Δ-polarization } τ ↦ ( ℂg/Λ, ℒ )$ is a bijection.
2. $\mathrm{dim}\left({H}^{0}\left({ℂ}^{g}/\Lambda ,ℒ\right)\right)={d}_{1}{d}_{2}\cdots {d}_{g}.$

Notes and References

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

The $\mathrm{Im}\tau >0$ condition in ${𝒢}_{g}$ is what is providing the positive definiteness of $⟨\phantom{A},\phantom{A}⟩$ (and hence the ampleness of $ℒ$).

The action of $\mathrm{Sp}\left(\Delta ,ℤ\right)$ on ${𝒢}_{g}$ given in (AbV 2) is a generalization of the action of ${\mathrm{Sp}}_{2}\left(ℤ\right)={\mathrm{SL}}_{2}\left(ℤ\right)$ on the upper half plane by Möbius transformations.

The integral symplectic group of degree $g$, or Siegel modular group of degree $g$ is ${\mathrm{Sp}}_{2g}\left(ℤ\right)=\mathrm{Sp}\left(\mathrm{Id},ℤ\right).$

This section follows [SU, §2.1].

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

References

[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.