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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

See also

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

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