Extending the Torelli map to toroidal compactifications of Siegel space (Q421016)

Let \(A_g\) denote the moduli space of principally polarized abelian varieties of dimension \(g\). Let \(\tau^{\text{perf}}\) (resp. \(\tau^{\text{vor}}\), \(\tau^{\text{cent}}\)) denote the first Voronoi fan consisting of perfect cones (resp. the second Voronoi fan, central cones) and let \(\bar{A}_g^{\text{perf}}\) (resp. \(\bar{A}_g^{\text{vor}}\), \(\bar{A}_g^{\text{cent}}\)) be the toroidal compactification of \(A_g\) determined by the corresponding fan. For the second Voronoi fan, the Torelli map from the moduli space \(M_g\) of curves of genus \(g\) to \(A_g\) is proved by Mumford and Namikawa to extend to a regular map from the Deligne-Mumford compactification \(\bar{M}_g\) to \(\bar{A}_g^{\text{vor}}\) (see \textit{Y. Namikawa} [Math. Ann. 221, 97--141 (1976; Zbl 0306.14016)] and [Math. Ann. 221, 201--241 (1976; Zbl 0327.14013)]). In this paper the authors prove that the extended Torelli map to \(\bar{A}_g^{\text{perf}}\) is also regular. They also show that the map to \(\bar{A}_g^{\text{cent}}\) is not regular for \(g\geq 9\); this disproves a 1973 conjecture of Namikawa.