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Let $A_g$ be the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. (EDIT: I mean the coarse moduli space.) Is this smooth?

Since $A_g$ is the quotient of Siegel upper half space by $\mathrm{Sp}_{2g}(\mathbb{Z})$ and this group has torsion elements, it seems likely that the answer is no. On the other hand $\mathrm{SL}_2(\mathbb{Z})$ already has torsion elements and $A_1$ is smooth.

Any attempt I make to search for information on this question only leads to information about the smoothness of the boundaries in compactifications of $A_g$.

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I am curious \emph{why} you should like to know whether or not $A_g$ is smooth? Is there a particular result that you are trying to extend? Are you looking to describe deRham cohomology of $A_g$? –  J. Martel Apr 25 '13 at 20:33
I was writing an exposition of some of the theory of $A_g$. I wrote that the moduli space with level structure $A_{g,1,n}$ is smooth for $n \geq 3$, and I wondered whether the condition $n \geq 3$ is necessary. –  Martin Orr Apr 26 '13 at 11:21
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2 Answers

up vote 7 down vote accepted

The answer is no, it is not smooth for any $g \geq 2$. For $g \geq 3$ the singular locus is precisely the locus of PPAVs with automorphism group greater than $\pm \mathrm{id}$, as proven in Oort, Frans: "Singularities of coarse moduli schemes". For $g=2$ there is IIRC a unique singular point which is in $M_2$ (the open subvariety of $A_2$ of Jacobians), I think this is in Igusa's paper "Arithmetic variety of moduli for genus two".

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What is IIRC? And the image of the period locus in genus 2 is closed, c.f. Mess' "Torelli group of genus 2 and 3 surfaces". –  J. Martel Apr 25 '13 at 20:25
IIRC is internet slang for "if I recall correctly" and PPAVs is internet slang for "principally polarized abelian varieties". –  Ashwath Rabindranath Apr 26 '13 at 1:08
I think PPAV must be older than the internet. –  dhagbert Apr 26 '13 at 8:48
J. Martel, either I am misunderstanding you or you are confused. The locus of Jacobians is not closed in $A_g$ for any $g \geq 2$, its closure is the locus of products of Jacobians. When $g=2$ every ppav is a Jacobian or a product of two elliptic curves. –  Dan Petersen Apr 26 '13 at 10:29
The confusion is my own part--the jacobian locus is definitely not closed (and actually, very obviously so). –  J. Martel Apr 28 '13 at 17:07
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There is an ambiguity in the question, which lies of course in the definition of moduli space, as the functor defining $A_g$ is not representable in the category of schemes. One solution, which seems the one considered implicitly by the OP (as suggested by the claim that for $g=1$ the moduli space is $\mathbb A^1$), and by Dan Petersen in his answer, is to define $A_g$ as the coarse moduli space of PPAV of genus $g$. And in this case, indeed, $A_g$ is not smooth. An other solution, which has many advantaged, is to consider $A^g$ not as a scheme but as an algebraic stack. In many respect this is the right thing to do, and in this case then $A^g$ is smooth as an algebraic stack. A proof for this is in the book of Faltings-Chai, Degenerations of abelian varieties.

(PS: I believe that the OP is well-aware of this distinction coarse/fine moduli space, but since it was mentioned in the question nor in the first answer, it seemed important re recall it for other readers).

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I agree with all of this but it seems tough going to refer to Faltings--Chai for this result. The OP seems content to work over the complex numbers and then smoothness of the moduli stack amounts to saying that there exists a finite index subgroup of $\mathrm{Sp}(2g,\mathbf Z)$ which acts freely on Siegel space. –  Dan Petersen Apr 25 '13 at 14:42
You are quite right, I meant to ask about the coarse moduli space. –  Martin Orr Apr 25 '13 at 14:47
@Dan Petersen: One can prove that the analytic quotient is isomorphic to the analytification of the moduli scheme (with "enough" level) as complex-analytic spaces (not just as sets) by a general principle without assuming smoothness on the algebraic side (and thereby get an analytic proof of algebraic smoothness): use relative exp maps over possibly non-smooth analytic spaces to construct a map from the analytic quotient to the analytification of the moduli scheme and then get bijectivity on artinian points via GAGA to conclude. Or prove smoothness algebraically; cf. Oort's 1971 paper. :) –  user29283 Apr 25 '13 at 15:24
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