Moduli of abelian varieties

Abelian varieties are a natural generalization of elliptic curves to higher dimensions. However, unlike the case of elliptic curves, there is no well-behaved stack playing the role of a moduli stack for higher-dimensional abelian varieties. One can solve this problem by constructing a moduli stack of abelian varieties equipped with extra structure, such as a principal polarisation. Just as there is a moduli stack of elliptic curves over $ℂ$ constructed as a stacky quotient of the upper-half plane by the action of $S{L}_2(\mathbb{Z})$, there is a moduli space of principally polarised abelian varieties given as a stacky quotient of Siegel upper half-space by the symplectic group ${\mathrm{Sp}}_{2g}(\mathbb{Z})$. By adding even more extra structure, such as a level n structure, one can go further and obtain a fine moduli space.

Constructions over characteristic 0

Principally polarized Abelian varieties

Recall that the Siegel upper half-space

$H_g$

is the set of symmetric

$g\times g$

complex matrices whose imaginary part is positive definite. This an open subset in the space of

$g\times g$

symmetric matrices. Notice that if

$g=1$

,

$H_g$

consists of complex numbers with positive imaginary part, and is thus the upper half plane, which appears prominently in the study of elliptic curves. In general, any point

$\mathrm{\Omega }\in H_g$

gives a complex torus

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

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

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

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

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

which gives a Deligne-Mumford stack over

$\mathrm{Spec}(ℂ)$

. 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 principally polarized Abelian varieties equipped with level n-structure because this breaks the symmetries and gives a moduli space 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

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

where

$L$

is the lattice

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

. 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

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

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