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Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space \mathcal_ over characteristic 0 constructed as a quotient of the upper-half plane by the action of SL_2(\mathbb), there is an analogous construction for abelian varieties \mathcal_g using the Siegel upper half-space and the symplectic group \operatorname_(\mathbb).


Constructions over characteristic 0


Principally polarized Abelian varieties

Recall that the Siegel upper-half plane is given by
H_g = \ \subseteq \operatorname_g(\mathbb)
which is an open subset in the g\times g symmetric matrices (since \operatorname(\Omega) > 0 is an open subset of \mathbb, and \operatorname is continuous). Notice if 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^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 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.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 A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
, 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 In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesim ...
Abelian varieties Elliptic curves