A torsion property of the zero of Kodaira-Spencer over ${\mathbb{P}}^1$ removing four points

Abstract

We establish a torsion theorem to the effect that the unique zero of the Kodaira-Spencer map attached to a certain quasi-semistable family of complex projective varieties over the complex projective line is the image of a torsion point of an elliptic curve under the natural projection. The proof is a mod $p$ argument and requires a density one set of primes. There are three essential ingredients in the proof: a solution to the conjecture of Sun-Yang-Zuo, which constitutes the principal part of the paper, Pink's theorem, and Higgs periodicity theorem.

arXiv comment3rd version. Acknowledgement to the initial discovery of the torsion phenomenon due to Jingbang Yang and Kang Zuo as proved in the paper added