Torelli locus and rigidity
In the article under review, the author studies the extent to which a locally symmetric space can be realized as a subvariety of \(\mathcal{M}_g\), the moduli space of genus \(g \geq 2\) curves. The author's main result is that if \(M\) is a subvariety of \(\mathcal{M}_g\) which is immersed by the Torelli map as a subvariety \(N\) in the Siegel upper half space \(\mathfrak{S}_g\) and if \(N\) is not a real or complex hyperbolic space form, then \(N\) cannot be a totally geodesic subvariety with respect to the Bergman metric. This result establishes a special case of some related conjectures of Coleman and Oort, see for instance [\textit{F. Oort}, Symp. Math. 37, 228--234 (1997; Zbl 0911.14018)]. In the course of proving it, the author gives a unified proof of a superrigidity result on mapping class groups. This auxiliary result is of an independent interest and builds on work of \textit{G. Daskalopoulos} and \textit{C. Mese} [Invent. Math. 224, No. 3, 791--916 (2021; Zbl 1470.32032)], the author's work [Int. Math. Res. Not. 2003, No. 31, 1677--1686 (2003; Zbl 1037.57011)] in addition to earlier work of \textit{B. Farb} and \textit{H. Masur} [Topology 37, No. 6, 1169--1176 (1998; Zbl 0946.57018)].
scientific article; zbMATH DE number 7599286
scientific article