Moduli of abelian varieties explained
over characteristic 0 constructed as a quotient of the
upper-half plane by the action of
,
[1] there is an analogous construction for abelian varieties
using the
Siegel upper half-space and the
symplectic group
.
[2] Constructions over characteristic 0
Principally polarized Abelian varieties
Recall that the Siegel upper-half plane is given by[3]
Hg=\{\Omega\in\operatorname{Mat}g,g(C):\OmegaT=\Omega,\operatorname{Im}(\Omega)>0\}\subseteq\operatorname{Sym}g(C)
which is an open subset in the
symmetric matrices (since
\operatorname{Im}(\Omega)>0
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
X\Omega\congX\Omega'\iff\Omega=M\Omega'
for M\in\operatorname{Sp}2g(Z)
hence the moduli space of principally polarized abelian varieties is constructed from the
stack quotientl{A}g=[\operatorname{Sp}2g(Z)\backslashHg]
which 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.[4] 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(X\Omega,Z/n)\cong
⋅ L/L\congn-torsionofX\Omega
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
\Gamma(n)=\ker[\operatorname{Sp}2g(Z)\to\operatorname{Sp}2g(Z)/n]
and define
Ag,n=\Gamma(n)\backslashHg
as a quotient variety.
References
- Hain. Richard. 2014-03-25. Lectures on Moduli Spaces of Elliptic Curves. math.AG. 0812.1803.
- Web site: Arapura. Donu. Abelian Varieties and Moduli.
- Book: Birkenhake. Christina. Complex Abelian Varieties. Lange. Herbert. 2004. Springer-Verlag. 978-3-540-20488-6. 2. Grundlehren der mathematischen Wissenschaften. Berlin Heidelberg. 210–241. en.
- Level n-structures are used to construct an intersection theory of Deligne–Mumford stacks
See also