Mathematical Research Letters

Volume 29 (2022)

Number 2

The Shafarevich conjecture and some extension theorems for proper hyperbolic polycurves

Pages: 541 – 558



Ippei Nagamachi (Graduate School of Mathematical Sciences, University of Tokyo, Japan)

Teppei Takamatsu (Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan)


In this paper, we prove the Shafarevich conjecture for proper hyperbolic polycurves, which is a higher dimensional analogue of that for proper hyperbolic curves. We prove this by using the good reduction criterion for proper hyperbolic polycurves established in [19], which gives a new proof of the main theorem of [7] for canonically polarized surfaces smooth proper over curves. Our theorem is a generalization of the result of Javanpeykar [7] in the following two points: (i) We treat proper smooth models which are not necessarily canonically polarized. (ii) We treat higher dimensional varieties, that is, proper hyperbolic polycurves of any dimension. This paper also contains a generalization of the moduli theory of Kodaira fibrations due to Jost and Yau [13].

The first author was supported by Iwanami Fujukai Foundation. The second author would like to thank Qing Liu for helpful comments on Proposition 1.7. The second author is supported by the FMSP program at the University of Tokyo. This work was supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University. They thank the referee for giving a simple proof of Lemma 1.6.

Received 5 November 2020

Accepted 24 December 2020

Published 29 September 2022