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. 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 $\mathbb$ constructed as a stacky quotient of the upper-half plane by the action of $SL_2(\mathbb)$, there is a moduli space of principally polarised abelian varieties given as a stacky quotient of Siegel upper half-space by the symplectic group $\operatorname_(\mathbb)$. By adding even more extra structure, such as a level ''n'' structure, one can go further and obtain a fine moduli space.

Constructions over characteristic 0

Principally polarized Abelian varieties

Recall that the Siegel upper half-space $H_g$ is the set of symmetric $g \times g$ complex matrices whose imaginary part is positive definite. This an open subset in the space of $g\times g$ symmetric matrices. Notice that if $g=1$, $H_g$ 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 $\Omega \in H_g$ gives a complex torus

$X_\Omega = \mathbb^g/(\Omega\mathbb^g + \mathbb^g)$

with a principal polarization $H_\Omega$ from the matrix $\Omega^$^{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

$X_\Omega \cong X_ \iff \Omega = M\Omega'$ for $M \in \operatorname_(\mathbb)$

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

\mathcal_g = operatorname_(\mathbb)\backslash H_g/math>

which gives a Deligne-Mumford stack over $\operatorname(\mathbb)$. 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 principally polarized Abelian varieties equipped with level ''n''-structure because this breaks the symmetries and gives a moduli space instead of a moduli stack.Level ''n''-structures are used to construct an intersection theory of Deligne–Mumford stacks 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

: $H_1(X_\Omega, \mathbb/n) \cong \frac\cdot L/L \cong n\text X_\Omega$

where $L$ is the lattice $\Omega\mathbb^g + \mathbb^g \subset \mathbb^$. 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_(\mathbb) \to \operatorname_(\mathbb/n)/math>

and define

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

as a quotient variety.

References

See also

*Schottky problem
*Siegel modular variety
*Moduli stack of elliptic curves
*Moduli of algebraic curves
* Hilbert scheme
* Deformation Theory

Abelian varieties
Elliptic curves