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Constructions over characteristic 0

Principally polarized Abelian varieties

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

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

with a principal polarization H Ω {\displaystyle H_{\Omega }} from the matrix Ω − 1 {\displaystyle \Omega ^{-1}} ⁵page 34. It turns out all principally polarized Abelian varieties arise this way, giving H g {\displaystyle H_{g}} the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where

X Ω ≅ X Ω ′ ⟺ Ω = M Ω ′ {\displaystyle X_{\Omega }\cong X_{\Omega '}\iff \Omega =M\Omega '} for M ∈ Sp 2 g ⁡ ( Z ) {\displaystyle M\in \operatorname {Sp} _{2g}(\mathbb {Z} )}

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

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

which gives a Deligne-Mumford stack over Spec ⁡ ( C ) {\displaystyle \operatorname {Spec} (\mathbb {C} )} . If this is instead given by a GIT quotient, then it gives the coarse moduli space A g {\displaystyle 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.⁶⁷ 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 Ω , Z / n ) ≅ 1 n ⋅ L / L ≅ n -torsion of  X Ω {\displaystyle H_{1}(X_{\Omega },\mathbb {Z} /n)\cong {\frac {1}{n}}\cdot L/L\cong n{\text{-torsion of }}X_{\Omega }}

where L {\displaystyle L} is the lattice Ω Z g + Z g ⊂ C 2 g {\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

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

and define

A g , n = Γ ( n ) ∖ H g {\displaystyle A_{g,n}=\Gamma (n)\backslash 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

References

1. On the moduli stack of abelian varieties without polarization: https://mathoverflow.net/q/358411/2893 https://mathoverflow.net/q/358411/2893 ↩

2. Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG]. /wiki/ArXiv_(identifier) ↩

3. Arapura, Donu. "Abelian Varieties and Moduli" (PDF). https://www.math.purdue.edu/~arapura/preprints/abelian.pdf ↩

4. 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. 978-3-540-20488-6 ↩

5. Arapura, Donu. "Abelian Varieties and Moduli" (PDF). https://www.math.purdue.edu/~arapura/preprints/abelian.pdf ↩

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 978-1-4757-9286-7 978-1-4757-9286-7 ↩

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