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 ${\displaystyle SL_{2}(\mathbb {Z} )}$[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 ${\displaystyle \mathbb {R} }$, and ${\displaystyle \operatorname {Im} }$ is continuous). Notice if ${\displaystyle g=1}$ this gives ${\displaystyle 1\times 1}$ 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 ${\displaystyle \Omega ^{-1}}$[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 ${\displaystyle A_{g}}$.

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 ${\displaystyle L}$ 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

1. Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math].
2. Arapura, Donu. "Abelian Varieties and Moduli" (PDF).
3. 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.
4. 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, retrieved 2020-06-05
5. 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