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 $A_g$.

  • $\begingroup$ 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 $A_g$? $\endgroup$ – J. Martel Apr 25 '13 at 20:33
  • $\begingroup$ 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. $\endgroup$ – Martin Orr Apr 26 '13 at 11:21

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".

  • $\begingroup$ 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". $\endgroup$ – J. Martel Apr 25 '13 at 20:25
  • 1
    $\begingroup$ IIRC is internet slang for "if I recall correctly" and PPAVs is internet slang for "principally polarized abelian varieties". $\endgroup$ – Ashwath Rabindranath Apr 26 '13 at 1:08
  • 2
    $\begingroup$ I think PPAV must be older than the internet. $\endgroup$ – dhagbert Apr 26 '13 at 8:48
  • $\begingroup$ 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. $\endgroup$ – Dan Petersen Apr 26 '13 at 10:29
  • $\begingroup$ The confusion is my own part--the jacobian locus is definitely not closed (and actually, very obviously so). $\endgroup$ – J. Martel Apr 28 '13 at 17:07

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).

  • $\begingroup$ 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. $\endgroup$ – Dan Petersen Apr 25 '13 at 14:42
  • $\begingroup$ You are quite right, I meant to ask about the coarse moduli space. $\endgroup$ – Martin Orr Apr 25 '13 at 14:47
  • $\begingroup$ @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. :) $\endgroup$ – user29283 Apr 25 '13 at 15:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.