Abelian varieties are a natural generalization of
elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a
natural moduli space over characteristic 0 constructed as a quotient of the
upper-half plane by the action of
, there is an analogous construction for abelian varieties
using the
Siegel upper half-space and the
symplectic group .
Constructions over characteristic 0
Principally polarized Abelian varieties
Recall that the
Siegel upper-half plane is given by
which is an open subset in the
symmetric matrices (since
is an open subset of
, and
is continuous). Notice if
this gives
matrices with positive imaginary part, hence this set is a generalization of the upper half plane. Then any point
gives a complex torus
with a principal polarization
from the matrix
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
for
hence the moduli space of principally polarized abelian varieties is constructed from the
stack quotientwhich gives a
Deligne-Mumford stack over
. 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 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
:
where
is the lattice
. 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
and define
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