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arXiv · 2602.16877

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

Emmett Lennen
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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.

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Emmett Lennen. 2026-02-18. Dimension bounds for relative character varieties on the projective line with three punctures $G=GL(r), O(r), Sp(r)$. https://arxiv.org/abs/2602.16877

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