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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 $A_g^0$.

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What do you mean by "indecomposable" ? –  Damian Rössler Aug 14 '13 at 8:26
    
I mean not a product of lower-dimensional abelian varieties. They are allowed to be isogenous to such a product. –  Dan Petersen Aug 14 '13 at 8:50
    
Why is the property of being indecomposable an open condition ? This property is equivalent to the existence of an idempotent in the endomorphism algebra of the abelian variety and I don't see why this should be a constructible (let alone open) property (sorry if I missed an obvious point). –  Damian Rössler Aug 14 '13 at 21:56
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Being indecomposable is also equivalent to not being in the image of any of the finitely many "gluing" maps $A_h \times A_{g-h} \to A_g$. –  Dan Petersen Aug 15 '13 at 6:04
    
Ahh right, of course, thank you. –  Damian Rössler Aug 15 '13 at 7:36
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