Dimension bounds for relative character varieties on the projective line with three punctures $G=GL(r), O(r), Sp(r)$

TL;DR

This paper establishes explicit linear bounds on the rank of certain relative character varieties on the thrice-punctured projective line, using Simpson's diagrammatic method and Katz's middle convolution, with sharp bounds in specific cases.

Contribution

It provides the first explicit linear upper bounds on the rank of MC-minimal character varieties for $GL(r)$, $O(r)$, and $Sp(r)$, and shows these bounds are sharp in key cases.

Findings

Derived explicit linear bounds $R(d)$ for the rank $r$ of character varieties.

Proved that arbitrary character varieties are isomorphic to those satisfying the bounds via Katz's middle convolution.

Established the sharpness of bounds for $GL(r)$ and non-overlapping quadratic cases.

Abstract

We consider relative character varieties on $\mathbb{P}^1\backslash\{0,1,\infty\}$ with $G=GL(r), O(r)$, or $Sp(r)$. Using a diagrammatic method of Simpson's, we give an explicit linear upper bound $R(d)$ on the rank $r$ of an MC-minimal character variety of dimension $d>2$. An arbitrary character variety is isomorphic, via Katz's middle convolution, to one satisfying the bound. For the general linear and non-overlapping quadratic cases, the bounds we give are the sharpest possible using this method.