Abelian Varieties

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $$\Lambda =\mathbb{Z}-span\{a_1,a_2,...,a_{2g}\}with{a}_1,...,a_{2g}\in {ℂ}^g.$$ The period matrix of ${ℂ}^g/\Lambda$ is $$\Omega =\left(\begin{array}{ccc}-& a_1& -\\ -& a_2& -\\ & ⋮& \\ -& a_{2g}& -\end{array}\right)\in M_{2g\times g}\left(ℂ\right).$$ The matrix $\Omega$ is rank $2g$ (i.e. $\Lambda$ has rank $2g$) if and only if $\det (\Omega ,\stackrel{\_}{\Omega })\ne 0.$ Let $$ℳ=\{\Omega \in M_{2g\times g}\left(ℂ\right)|\det (\Omega ,\stackrel{\_}{\Omega })\ne 0\}.$$ Then $$\begin{array}{rcl}{\mathrm{GL}}_{2g}\backslash ℳ/{\mathrm{GL}}_g\left(ℂ\right)& \stackrel{\sim }{\to }& \left\{\genfrac{}{}{0ex}{}{isomorphism\; classes\; of}{g-dimensional\; complex\; tori}\right\}\\ \Omega & \mapsto & {ℂ}^g/\Lambda ,\end{array}where\Lambda =\mathbb{Z}-span\left(\Omega \right).$$ Each double coset in ${\mathrm{GL}}_{2g}\left(\mathbb{Z}\right)\backslash ℳ/{\mathrm{GL}}_g\left(ℂ\right)$ has a representative of the form $$\Omega =\left(\genfrac{}{}{0ex}{}{\tau }{I_g}\right)with\tau \in M_{g\times g}\left(ℂ\right)and\det \left(\mathrm{Im}\tau \right)\ne 0.$$ The action of ${\mathrm{GL}}_{2g}\left(\mathbb{Z}\right)$ on coset representatives is $$M\cdot \tau =(A\tau +B){(C\tau +D)}^{-1}forM=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in {\mathrm{GL}}_{2g}\left(\mathbb{Z}\right).$$

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 $A\&hkrarr;{\mathbb{P}}^d\left(ℂ\right).$

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

In this case $$A\hookrightarrow \mathbb{P}\left(V\right)whereA=H^0(A,ℒ).$$

(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. $$ℒ={ℂ}^g{\times }_{\Lambda }ℂ=\frac{{ℂ}^g\times ℂ}{⟨(z,c)=(z+\mu ,\chi \left(\mu \right)e^{\pi (⟨z,\mu ⟩+\frac{1}{2}⟨\mu ,\mu ⟩)}c)for\mu \in \Lambda ⟩}$$ for a unique Hermitian form $$⟨\phantom{A},\phantom{A}⟩:{ℂ}^g\times {ℂ}^g\to ℂand\; function\chi :\Lambda \to U_1\left(ℂ\right)$$ satisfying
   1. if $\mu ,\nu \in \Lambda$ then
   2. $\mathrm{Im}\left(⟨\mu ,\nu ⟩\right)\in \mathbb{Z}$ and $\chi (\mu +\nu )=\chi \left(\mu \right)\chi \left(\nu \right)e^{i\pi \mathrm{Im}\left(⟨\mu ,\nu ⟩\right)}.$
2. The line bundle $ℒ$ is ample if and only if $$⟨\phantom{A},\phantom{A}⟩:{ℂ}^g\times {ℂ}^g\to ℂ$$ is positive definite.

The Siegel upper half space is $${𝒢}_g=\{\tau \in M_{g\times g}\left(ℂ\right)|{\tau }^t=\tau ,\mathrm{Im}\tau >0\}.$$ 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 $d_1,...,d_g\in {\mathbb{Z}}_{>0}$ with $d_1\left|d_2\right|\cdots |d_g$ so that $$\begin{array}{ll}\Lambda =\mathbb{Z}-span\{a_1,...,a_{2g}\}and⟨z,w⟩=z{\left(\mathrm{Im}\tau \right)}^{-1}{\stackrel{\_}{w}}^t& (AbV\; 1)\end{array}$$ where, in the basis $\frac{1}{d_1}{\gamma }_{g+1}, \frac{1}{d_2}{\gamma }_{g+2}, ..., \frac{1}{d_g}{\gamma }_{2g}$ of ${ℂ}^g$ $$\Omega =\left(\begin{array}{ccc}-& a_1& -\\ -& a_2& -\\ & ⋮& \\ -& a_{2g}& -\end{array}\right)=\left(\genfrac{}{}{0ex}{}{\tau }{\Delta }\right)with\; \tau \in {𝒢}_g \; and\; \Delta =\left(\begin{array}{ccc}{d}_1& & 0\\ & \ddots & \\ 0& & d_g\end{array}\right).$$ The freedom in the choice of the basis ${\gamma }_1,...,{\gamma }_{2g}$ is controlled by the paramodular group $$\mathrm{Sp}(\Delta ,\mathbb{Z})=\{M\in {\mathrm{GL}}_{2g}\left(\mathbb{Z}\right)|M\left(\begin{array}{cc}0& \Delta \\ -\Delta & 0\end{array}\right)M^t=\left(\begin{array}{cc}0& \Delta \\ -\Delta & 0\end{array}\right)\}.$$ Define an action of $\mathrm{Sp}(\Delta ,\mathbb{Z})$ on ${𝒢}_g$ by $$\begin{array}{ll}M\cdot \tau =(\alpha \tau +\beta \Delta ){(\gamma \tau +\delta \Delta )}^{-1}\Delta ,ifM=\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\in \mathrm{Sp}(\Delta ,\mathbb{Z}).& (AbV\; 2)\end{array}$$

Let $d_1,...,d_g\in {\mathbb{Z}}_{>0}$ with $d_1\left|d_2\right|\cdots |d_g.$ 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 $$\begin{array}{rcl}\mathrm{Sp}(\Delta ,\mathbb{Z})\backslash {𝒢}_g& \to & \left\{\genfrac{}{}{0ex}{}{\genfrac{}{}{0ex}{}{isomorphism\; classes}{of\; abelian\; varieties}}{with\Delta -polarization}\right\}\\ \tau & \mapsto & ({ℂ}^g/\Lambda ,ℒ)\end{array}$$ is a bijection.
2. $\dim \left(H^0({ℂ}^g/\Lambda ,ℒ)\right)=d_1{d}_2\cdots d_g.$

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 $\mathrm{Sp}(\Delta ,\mathbb{Z})$ on ${𝒢}_g$ given in (AbV 2) is a generalization of the action of ${\mathrm{Sp}}_2\left(\mathbb{Z}\right)={\mathrm{SL}}_2\left(\mathbb{Z}\right)$ on the upper half plane by Möbius transformations.

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

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.

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