Moduli of abelian varieties

Abelian varieties are a natural generalization of elliptic curves to higher dimensions. However, unlike the case of elliptic curves, there is no well-behaved stack playing the role of a moduli stack for higher-dimensional abelian varieties.[1] One can solve this problem by constructing a moduli stack of abelian varieties equipped with extra structure, such as a principal polarisation. Just as there is a moduli stack of elliptic curves over ℂ constructed as a stacky quotient of the upper-half plane by the action of SL2(ℤ),[2] there is a moduli space of principally polarised abelian varieties given as a stacky quotient of Siegel upper half-space by the symplectic group Sp2g(ℤ).[3] By adding even more extra structure, such as a level n structure, one can go further and obtain a fine moduli space.

## Contents

* 1 Constructions over characteristic 0
* 1.1 Principally polarized Abelian varieties
* 1.2 Principally polarized Abelian varieties with level n structure
* 2 References
* 3 See also

## Constructions over characteristic 0

### Principally polarized Abelian varieties

Recall that the Siegel upper half-space

is the set of symmetric

g×g

complex matrices whose imaginary part is positive definite.[4] This an open subset in the space of

g×g

symmetric matrices. Notice that if

g=1

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∈Hg

gives a complex torus

> XΩ=ℂg/(Ωℤg+ℤg)

with a principal polarization

from the matrix

Ω−1

[3]page 34. It turns out all principally polarized Abelian varieties arise this way, giving

the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where

> XΩ≅XΩ′⟺Ω=MΩ′

> for

> M∈Sp2g(ℤ)

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

> 𝒜g=[Sp2g(ℤ)∖Hg]

which gives a Deligne-Mumford stack over

Spec⁡(ℂ)

. If this is instead given by a GIT quotient, then it gives the coarse moduli space

### 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.[5][6] 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

H1(XΩ,ℤ/n)≅1n⋅L/L≅n-torsion of XΩ

where

is the lattice

Ωℤg+ℤg⊂ℂ2g

. 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⁡[Sp2g(ℤ)->Sp2g(ℤ/n)]

and define

> Ag,n=Γ(n)∖Hg

as a quotient variety.

## References

1. ↑ On the moduli stack of abelian varieties without polarization: https://mathoverflow.net/q/358411/2893
2. ↑ Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG].
3. ↑ 3.0 3.1 Arapura, Donu. "Abelian Varieties and Moduli". https://www.math.purdue.edu/~arapura/preprints/abelian.pdf.
4. ↑ Birkenhake, Christina; Lange, Herbert (2004) (in en). Complex Abelian Varieties. Grundlehren der mathematischen Wissenschaften (2 ed.). Berlin Heidelberg: Springer-Verlag. pp. 210–241. ISBN 978-3-540-20488-6. https://www.springer.com/gp/book/9783540204886.
5. ↑ Mumford, David (1983), Artin, Michael; Tate, John, eds., "Towards an Enumerative Geometry of the Moduli Space of Curves" (in en), 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
6. ↑ 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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