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 constructed as a stacky quotient of the upper-half plane by the action of ,[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 .[3] 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
[edit]Principally polarized Abelian varieties
[edit]Recall that the Siegel upper half-space is the set of symmetric complex matrices whose imaginary part is positive definite.[4] This an open subset in the space of symmetric matrices. Notice that if , 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 gives a complex torus
with a principal polarization from the matrix [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
for
hence the moduli space of principally polarized abelian varieties is constructed from the stack quotient
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
[edit]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.[5][6] 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
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
[edit]- ^ On the moduli stack of abelian varieties without polarization: https://mathoverflow.net/q/358411/2893
- ^ Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG].
- ^ a b Arapura, Donu. "Abelian Varieties and Moduli" (PDF).
- ^ Birkenhake, Christina; Lange, Herbert (2004). Complex Abelian Varieties. Grundlehren der mathematischen Wissenschaften (2 ed.). Berlin Heidelberg: Springer-Verlag. pp. 210–241. ISBN 978-3-540-20488-6.
- ^ Mumford, David (1983), Artin, Michael; Tate, John (eds.), "Towards an Enumerative Geometry of the Moduli Space of Curves", 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
- ^ Level n-structures are used to construct an intersection theory of Deligne–Mumford stacks