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 ${\mathcal {M}}_{1,1}$ over characteristic 0 constructed as a quotient of the upper-half plane by the action of $SL_{2}(\mathbb {Z} )$, there is an analogous construction for abelian varieties ${\mathcal {A}}_{g}$ using the Siegel upper half-space and the symplectic group $\operatorname {Sp} _{2g}(\mathbb {Z} )$.

Constructions over characteristic 0

Principally polarized Abelian varieties

Recall that the Siegel upper-half plane is given by

$H_{g}=\{\Omega \in \operatorname {Mat} _{g,g}(\mathbb {C} ):\Omega ^{T}=\Omega ,\operatorname {Im} (\Omega )>0\}\subseteq \operatorname {Sym} _{g}(\mathbb {C} )$

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

$X_{\Omega }=\mathbb {C} ^{g}/(\Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g})$

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

$X_{\Omega }\cong X_{\Omega '}\iff \Omega =M\Omega '$ for $M\in \operatorname {Sp} _{2g}(\mathbb {Z} )$

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

${\mathcal {A}}_{g}=[\operatorname {Sp} _{2g}(\mathbb {Z} )\backslash H_{g}]$

which gives a Deligne-Mumford stack over $\operatorname {Spec} (\mathbb {C} )$. If this is instead given by a GIT quotient, then it gives the coarse moduli space $A_{g}$.

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

$H_{1}(X_{\Omega },\mathbb {Z} /n)\cong {\frac {1}{n}}\cdot L/L\cong n{\text{-torsion of }}X_{\Omega }$

where $L$ is the lattice $\Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g}\subset \mathbb {C} ^{2g}$. 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

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

and define

$A_{g,n}=\Gamma (n)\backslash H_{g}$

as a quotient variety.

See also

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