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

TL;DR

This paper proves a torsion property of the zero of the Kodaira-Spencer map for certain families of complex varieties over the projective line, using a mod p approach and key theorems in algebraic geometry.

Contribution

It establishes a torsion theorem linking the zero of the Kodaira-Spencer map to torsion points on elliptic curves, solving a conjecture of Sun-Yang-Zuo.

Findings

Zero of Kodaira-Spencer map corresponds to torsion points

Solution to Sun-Yang-Zuo conjecture

Uses mod p techniques and key algebraic geometry theorems

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.