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).

arXiv comment17 pages, comments welcome

References in corpus (2)

• Surface group representations in ${\mathrm{SL}}_2(\mathbb{C})$ with finite mapping class orbits
• Pullback and direct image of parabolic connections and parabolic Higgs bundles