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

Constructions over characteristic 0
Principally polarized Abelian varieties

Recall that the Siegel upper-half plane is given by[3]

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

which is an open subset in the \( {\displaystyle g\times g} \) symmetric matrices (since \( {\displaystyle \operatorname {Im} (\Omega )>0} \) is an open subset of \( \mathbb {R} \) , and Im {\displaystyle \operatorname {Im} } \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 \( {\displaystyle \Omega \in H_{g}} \) gives a complex torus

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

with a principal polarization \( {\displaystyle H_{\Omega }} \) from the matrix \( \Omega ^{{-1}} \) [2]page 34. It turns out all principally polarized Abelian varieties arise this way, giving \( {\displaystyle H_{g}} \) the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where

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

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

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

which gives a Deligne-Mumford stack over \( {\displaystyle \operatorname {Spec} (\mathbb {C} )}. \) 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.[4][5] 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

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

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

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

and define

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

as a quotient variety.
References

Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG].
Arapura, Donu. "Abelian Varieties and Moduli" (PDF).
Birkenhake, Christina; Lange, Herbert (2004). Complex Abelian Varieties. Grundlehren der mathematischen Wissenschaften (2 ed.). Berlin Heidelberg: Springer-Verlag. pp. 210–241. ISBN 978-3-540-20488-6.
Mumford, David (1983), Artin, Michael; Tate, John (eds.), "Towards an Enumerative Geometry of the Moduli Space of Curves", Arithmetic and Geometry: Papers Dedicated to I.R. Shafarevich on the Occasion of His Sixtieth Birthday. Volume II: Geometry, Progress in Mathematics, Birkhäuser, pp. 271–328, doi:10.1007/978-1-4757-9286-7_12, ISBN 978-1-4757-9286-7

Level n-structures are used to construct an intersection theory of Deligne–Mumford stacks