Moduli of abelian varieties

Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space [math]\displaystyle{ \mathcal{M}_{1,1} }[/math] over characteristic 0 constructed as a quotient of the upper-half plane by the action of [math]\displaystyle{ SL_2(\mathbb{Z}) }[/math], there is an analogous construction for abelian varieties [math]\displaystyle{ \mathcal{A}_g }[/math] using the Siegel upper half-space and the symplectic group [math]\displaystyle{ \operatorname{Sp}_{2g}(\mathbb{Z}) }[/math].

Constructions over characteristic 0

Principally polarized Abelian varieties

Recall that the Siegel upper-half plane is given by

[math]\displaystyle{ H_g = \{ \Omega \in \operatorname{Mat}_{g,g}(\mathbb{C}) : \Omega^T =\Omega, \operatorname{Im}(\Omega) \gt 0 \} \subseteq \operatorname{Sym}_g(\mathbb{C}) }[/math]

which is an open subset in the [math]\displaystyle{ g\times g }[/math] symmetric matrices (since [math]\displaystyle{ \operatorname{Im}(\Omega) \gt 0 }[/math] is an open subset of [math]\displaystyle{ \mathbb{R} }[/math], and [math]\displaystyle{ \operatorname{Im} }[/math] is continuous). Notice if [math]\displaystyle{ g=1 }[/math] this gives [math]\displaystyle{ 1\times 1 }[/math] matrices with positive imaginary part, hence this set is a generalization of the upper half plane. Then any point [math]\displaystyle{ \Omega \in H_g }[/math] gives a complex torus

[math]\displaystyle{ X_\Omega = \mathbb{C}^g/(\Omega\mathbb{Z}^g + \mathbb{Z}^g) }[/math]

with a principal polarization [math]\displaystyle{ H_\Omega }[/math] from the matrix [math]\displaystyle{ \Omega^{-1} }[/math]^{page 34}. It turns out all principally polarized Abelian varieties arise this way, giving [math]\displaystyle{ H_g }[/math] the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where

[math]\displaystyle{ X_\Omega \cong X_{\Omega'} \iff \Omega = M\Omega' }[/math] for [math]\displaystyle{ M \in \operatorname{Sp}_{2g}(\mathbb{Z}) }[/math]

hence the moduli space of principally polarized abelian varieties is constructed from the stack quotient

[math]\displaystyle{ \mathcal{A}_g = [\operatorname{Sp}_{2g}(\mathbb{Z})\backslash H_g] }[/math]

which gives a Deligne-Mumford stack over [math]\displaystyle{ \operatorname{Spec}(\mathbb{C}) }[/math]. If this is instead given by a GIT quotient, then it gives the coarse moduli space [math]\displaystyle{ A_g }[/math].

Principally polarized Abelian varieties with level n-structure

In many cases, it is easier to work with the moduli space of principally polarized Abelian varieties with level n-structure because it creates a rigidification of the moduli problem which gives a moduli functor instead of a moduli stack. This means the functor is representable by an algebraic manifold, such as a variety or scheme, instead of a stack. A level n-structure is given by a fixed basis of

[math]\displaystyle{ H_1(X_\Omega, \mathbb{Z}/n) \cong \frac{1}{n}\cdot L/L \cong n\text{-torsion of } X_\Omega }[/math]

where [math]\displaystyle{ L }[/math] is the lattice [math]\displaystyle{ \Omega\mathbb{Z}^g + \mathbb{Z}^g \subset \mathbb{C}^{2g} }[/math]. Fixing such a basis removes the automorphisms of an abelian variety at a point in the moduli space, hence there exists a bona-fide algebraic manifold without a stabilizer structure. Denote

[math]\displaystyle{ \Gamma(n) = \ker [\operatorname{Sp}_{2g}(\mathbb{Z}) \to \operatorname{Sp}_{2g}(\mathbb{Z})/n] }[/math]

and define

[math]\displaystyle{ A_{g,n} = \Gamma(n)\backslash H_g }[/math]

as a quotient variety.

See also

• Schottky problem
• Siegel modular variety
• Moduli stack of elliptic curves
• Moduli of algebraic curves
• Hilbert scheme
• Deformation Theory