# Conceptual explanation for $\chi(\mathcal{A_g})=\chi(\mathcal M_{1} \times … \times \mathcal M_{g})$?

Let $$\mathcal{A_g}$$ be the moduli space of principally polarized abelian varieties over $$\mathbb C$$, $$\mathcal{M_g}$$ be the moduli space of smooth projective curves of genus $$g$$ over $$\mathbb C$$, and we regard both as Deligne-Mumford stacks or orbifolds. By computations we know their Euler characteristics :

$$\chi(\mathcal{A_g})=\chi(\mathcal M_{1})\chi(\mathcal M_{2})...\chi(\mathcal M_{g})=\prod_{k=1}^g \zeta(1-2k)$$.

Is there a conceptual explanation of this using Torelli map? Can we update it to an identity of Poincare polynomials?

• These Euler characteristics are not the alternating sums of Betti numbers and therefore aren't related to Poincare polynomials. For instance, they aren't integers. – Will Sawin Apr 17 at 2:12
• @WillSawin There are Poincaré polynomials associated with orbifolds. – sawdada Apr 17 at 2:53
• @WillSawin More precisely, one can replace $(-1)$ in $\chi(M/G) = \frac1{|G|}\sum_i\sum_{g\in G} (-1)^i\mathrm{Tr}_{H^i(M)}(g^*)$ by $t$ as in mathoverflow.net/questions/51993/… – sawdada Apr 17 at 14:25
• But that formula is for the Euler characteristic as an alternating sum of Betti numbers (see step 2 of Johannes Ebert's answer), not the other definition of the Euler characteristic for orbifolds, which gives $\chi(M/G) =\chi(M)/|G|$ and which is used in the zeta function identities you state. – Will Sawin Apr 17 at 14:50