将Torelli映射推广到Siegel空间的环面紧化
Extending Torelli map to toroidal compactifications of Siegel space
摘要：
自20世纪70年代以来，人们已经知道Torelli映射$M_g \to A_g$，将其雅可比矩阵与平滑曲线相关联，扩展到从Deligne-Mumford紧化$\bar{M}_g$到第二Voronoi紧化$\bar{A}_g^{vor}$的正则映射。证明了到完全锥（1st Voronoi）紧化$\bar{A}_g^{perf}$的广义Torelli映射也是正则的，而且$\bar{A}_g^{vor}$和$\bar{A}_g^{perf}$共享$\bar{M}_g$的象的一个公共Zerkki开邻域.我们还表明，映射到Igusa monoidal变换（中心锥紧化）是不规则的$g\ge9$，这反驳了1973年的Namikawa猜想。
英文摘要：
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 $g\ge9$; this disproves a 1973 conjecture of Namikawa.
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