Geometric local systems on the projective line minus four points

TL;DR

This paper explicitly constructs rank 2 local systems of geometric origin on the four-punctured projective line with specified monodromies, using Katz's middle convolution, and proves two related conjectures.

Contribution

It provides explicit constructions of certain local systems with prescribed monodromy, confirming two conjectures in the field.

Findings

Explicit construction of local systems with given monodromy

Application of Katz's middle convolution to geometric local systems

Proof of two conjectures by Sun-Yang-Zuo

Abstract

Let $J(m)$ be an $m\times m$ Jordan block with eigenvalue $1$. For $\lambda\in \mathbb{C}\setminus\{0,1\}$, we explicitly construct all rank $2$ local systems of geometric origin on $\mathbb{P}^1\setminus\{0,1,\lambda, \infty\}$, with local monodromy conjugate to $J(2)$ at $0,1,\lambda$ and conjugate to $-J(2)$ at $\infty$. The construction relies on Katz's middle convolution operation. We use our construction to prove two conjectures of Sun-Yang-Zuo (one of which was proven earlier by Lin-Sheng-Wang; the other was proven independently from us by Yang-Zuo).