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.^[1] 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 ${\displaystyle \mathbb {C} }${\displaystyle \mathbb {C} } constructed as a stacky quotient of the upper-half plane by the action of ${\displaystyle SL_{2}(\mathbb {Z} )}${\displaystyle SL_{2}(\mathbb {Z} )},^[2] there is a moduli space of principally polarised abelian varieties given as a stacky quotient of Siegel upper half-space by the symplectic group ${\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}${\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}.^[3] By adding even more extra structure, such as a level n structure, one can go further and obtain a fine moduli space.

Recall that the Siegel upper half-space ${\displaystyle H_{g}}${\displaystyle H_{g}} is the set of symmetric ${\displaystyle g\times g}${\displaystyle g\times g} complex matrices whose imaginary part is positive definite.^[4] This an open subset in the space of ${\displaystyle g\times g}${\displaystyle g\times g} symmetric matrices. Notice that if ${\displaystyle g=1}${\displaystyle g=1}, ${\displaystyle H_{g}}${\displaystyle 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 ${\displaystyle \Omega \in H_{g}}${\displaystyle \Omega \in H_{g}} gives a complex torus

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

with a principal polarization ${\displaystyle H_{\Omega }}${\displaystyle H_{\Omega }} from the matrix ${\displaystyle \Omega ^{-1}}${\displaystyle \Omega ^{-1}}^{[3] page 34}. It turns out all principally polarized Abelian varieties arise this way, giving ${\displaystyle H_{g}}${\displaystyle H_{g}} the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where

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

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

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

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

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.^{[5] [6]} 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

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

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

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

and define

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

as a quotient variety.

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

• Abelian varieties
• Elliptic curves