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Torelli locus and rigidity. (English) Zbl 1499.14050

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 [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 G. Daskalopoulos and 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 B. Farb and H. Masur [Topology 37, No. 6, 1169–1176 (1998; Zbl 0946.57018)].

MSC:

14H15 Families, moduli of curves (analytic)
14K10 Algebraic moduli of abelian varieties, classification
53C24 Rigidity results
53C43 Differential geometric aspects of harmonic maps
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