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 the complex numbers
* 1.1 Principally polarized Abelian varieties
* 1.2 Principally polarized Abelian varieties with level n structure
* 2 References
* 3 See also

## Constructions over the complex numbers

### 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, giving a moduli scheme instead of a moduli stack.[5][6] In other words, the functor that associates to a test scheme T the set of morphisms X -> T where all geometric fibers are principally polarized abelian varieties with level n-structure (intuitively, a family of objects parameterized by T) is actually representable by a scheme. 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

. An automorphism of an abelian variety with level structure is an automorphism of the variety that fixes this basis, and there are no such automorphisms other than the identity. Since no points have a stabilizer, the moduli space of principally polarized abelian varieties with level n-structure, which is a priori a stack, is actually a scheme. In this situation, where a family over arbitrary T is the same as a morphism from T into some scheme M (better: arises from a universal family through pullback), we say M is a fine moduli space. 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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