Moduli of abelian varieties

Last updated September 28, 2023

Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\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 ${\mathcal {A}}_{g}$ using the Siegel upper half-space and the symplectic group $\operatorname {Sp} _{2g}(\mathbb {Z} )$ .^[2]

Constructions over characteristic 0

Principally polarized Abelian varieties

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

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

which is an open subset in the $g\times g$ symmetric matrices (since $\operatorname {Im} (\Omega )>0$ is an open subset of $\mathbb {R}$ , and $\operatorname {Im}$ is continuous). Notice if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): 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 $\Omega \in H_{g}$ gives a complex torus

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

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

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

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

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

which gives a Deligne-Mumford stack over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\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

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 $\Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g}\subset \mathbb {C} ^{2g}$ . 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

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

and define

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

as a quotient variety.

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References

↑ Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv: 0812.1803 [math.AG].
1 2 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