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 ${{ℳ}}_{1,1}$ over characteristic 0 constructed as a quotient of the upper-half plane by the action of $S{L}_2(\mathbb{\mathbb{Z}})$, there is an analogous construction for abelian varieties ${{𝒜}}_g$ using the Siegel upper half-space and the symplectic group ${\mathrm{Sp}}_{2g}(\mathbb{\mathbb{Z}})$.

Constructions over characteristic 0

Principally polarized Abelian varieties

Recall that the Siegel upper-half plane is given by

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

which is an open subset in the

$g\times g$

symmetric matrices (since

$\mathrm{Im}(\mathrm{\Omega })>0$

is an open subset of

$\mathbb{ℝ}$

, and

$\mathrm{Im}$

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

$\mathrm{\Omega }\in H_g$

gives a complex torus

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

with a principal polarization

$H_{\mathrm{\Omega }}$

from the matrix

${\mathrm{\Omega }}^{-1}$

^{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

$X_{\mathrm{\Omega }}\cong X_{{\mathrm{\Omega }}^{\prime }}⟺\mathrm{\Omega }=M{\mathrm{\Omega }}^{\prime }$

for

$M\in {\mathrm{Sp}}_{2g}(\mathbb{\mathbb{Z}})$

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

${{𝒜}}_g=[{\mathrm{Sp}}_{2g}(\mathbb{\mathbb{Z}})\mathrm{\setminus }{H}_g]$

which gives a Deligne-Mumford stack over

$\mathrm{Spec}(\mathbb{ℂ})$

. 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

$H_1(X_{\mathrm{\Omega }},\mathbb{\mathbb{Z}}/n)\cong \frac{1}{n}\cdot L/L\cong n\text{-torsion of }{X}_{\mathrm{\Omega }}$

where

$L$

is the lattice

$\mathrm{\Omega }{\mathbb{\mathbb{Z}}}^g+{\mathbb{\mathbb{Z}}}^g\subset {\mathbb{ℂ}}^{2g}$

. 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

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

and define

$A_{g,n}=\mathrm{\Gamma }(n)\mathrm{\setminus }{H}_g$

as a quotient variety.

See also

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