Geometric local systems on the projective line minus four points

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).