# On the moduli stack of abelian varieties without polarization

(I am especially interested in abelian surfaces and characteristic 0).

1. How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-Mumford) stack?

2. How bad is the is the stack of abelian varieties with full 2 level structure (so with a basis for $$A[2]$$)?

3. Consider the maps from either 2) above or stack of principally polarized abelian varieties to 1) above. Are these maps smooth, are the geometric fibers finite (ie, are there only finitely many principal polarizations on an abelian variety)?

Neither moduli space is a stack because every point has the automorphism $$-1$$ but the same is true for the moduli stack of elliptic curves and that is still a DM stack and not too bad.

Even in characteristic 0, the CM locus is higher dimensional so the "especially stacky" locus has high dimension but I don't know how serious the problem is.

For the second question, while $$-1$$ fixes the two level structure, I suppose a generic CM automorphism doesn't fix it so perhaps the second stack is very nice, or atleast almost as nice as that of elliptic curves?

• I don't even know how to prove that the prestack of abelian varieties is a stack, i.e., satisfies descent. So before you ask whether this is a reasonable algebraic stack' it might be a good idea to prove that it is a stack at all. For example the prestack of genus 1 curves (so elliptic curves without a fixed point) is not a stack. – Pol van Hoften Apr 24 at 20:17
• I agree, I was primarily thinking about the level 2 structure version (which I do think is a stack?). Anyway, it looks like none of these ideas work anyway. – Asvin Apr 24 at 20:20
• Why would adding a full level $2 structure help with descent? – Pol van Hoften Apr 24 at 20:22 • Now that I think about it, probably not. The point with elliiptic curves is that picking a point gives an ample divisor. I guess one would have to stackify in both cases. – Asvin Apr 24 at 20:58 • Thinking about it a little bit more, I wonder if an argument along the following lines works: Define a prestack that takes a test scheme T to the groupoid of isomorphism classes of abelian algebraic spaces', that is smooth proper group algebraic spaces over$T$, with connected geometric fibers. This prestack is certainly a stack, by descent for algebraic spaces and because all the above properties can be checked (fpqc) locally. But it is a theorem of Raynaud (c.f. Theorem 1.9 of Chai-Faltings) that every abelian algebraic space is actually isomorphic to an abelian scheme. – Pol van Hoften Apr 25 at 8:29 ## 1 Answer First, when defining the stack you will have the issue that there are formal deformations of abelian varieties which do not extend to families of abelian varieties over any reduced scheme. These are the deformations that do not respect any polarization. (In the complex analytic world these correspond to deformations of complex tori) So unless you have some very strange definition of the functor, the local structure of this stack will be at least as bad as the formal limit $$\lim_{n\to \infty} \operatorname{Spec} k[x]/x^n$$. I think this rules out ever having a smooth morphism from a scheme, and thus rules out being an Artin stack. For $$E$$ a non-CM elliptic curve, the automorphism group of $$E^n$$ is $$GL_n(\mathbb Z)$$. This shows that for $$n>1$$, points of this moduli stack can have infinitely many automorphisms. In particular, the diagonal is not quasicompact. Level $$2$$ structure doesn't help with this at all, you just get the group of $$n\times n$$ matrices congruent to $$1$$ mod $$2$$. The map from the stack of principally polarized abelian varieties to this stack is not smooth because deformations can kill a principal polarization, and the fibers are not finite, again because of examples like $$E^n$$, whose principal polarizations are in bijection with $$n \times n$$ symmetric positive definite integer matrices with determinant $$1$$. In summary: There's a reason you haven't heard of this stack before. P.S. You shouldn't worry so much about automorphisms of the generic point, which almost never cause problems in practice. It's everything else that you should worry about! • Thanks, that's very helpful! Is the stack with full level structure also not an artin stack? This might be a different question and it's a little off the cuff but how about if I consider the stack of abelian surfaces in char 0 with 1 geometric non torsion point with finite stabilizer group? – Asvin Apr 24 at 13:54 • @Asvin (1) The third paragraph of my answer discusses full level$2$structure. (2) The stack of abelian surfaces with one geometric non-torsion point with finite stabilizer is not even a stack, because these conditions are not reasonable conditions in a family of points. Consider which points in$E \times E\$ have this property - it's the complement of a union of infinitely many curves. – Will Sawin Apr 24 at 14:51
• Yes, I see now. Thanks again! – Asvin Apr 24 at 14:59