Let $A_g$ be the moduli space of principally polarized abelian varieties and $A_g^0$ the open substack of indecomposable ones. Abstractly we know $A_g^0$ has a compactification with complement a normal crossing divisor. Is there one with a modular interpretation?

*Motivation:* the Torelli map $M_g \to A_g$ lands inside $A_g^0$. The map $M_g \to A_g^0$ is a closed immersion on coarse spaces and a ramified double cover of its image in the sense of stacks; this is the Torelli theorem. One can also map the space of curves of compact type into $A_g$, or the whole Deligne-Mumford compactification into the second Voronoi compactification of $A_g$, but now the Torelli theorem fails badly. I wonder if the Torelli theorem can be made to hold on some other compactification of $A_g^0$.

together with the polarisationin the definition of decomposability (ie consider the product polarisations). $\endgroup$ – Damian Rössler Aug 17 '13 at 22:07