Extending Torelli map to toroidal compactifications of Siegel space

Abstract

It has been known since the 1970s that the Torelli map $M_g\to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification ${\bar{M}}_g$ to the 2nd Voronoi compactification ${\bar{A}}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification ${\bar{A}}_g^{perf}$ is also regular, and moreover ${\bar{A}}_g^{vor}$ and ${\bar{A}}_g^{perf}$ share a common Zariski open neighborhood of the image of ${\bar{M}}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regular for $g\ge 9$; this disproves a 1973 conjecture of Namikawa.

To appear in Inventiones Mathematicae

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