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

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.

Figures (7)

• Figure 1: Example of a diagram associated to a $GL(r)$ relative character variety.
• Figure 2: Diagram for Example \ref{['ex_gl_dim']}.
• Figure 3: Diagram for the $GL(r)$ configurations that achieve minimal dimension.
• Figure 4: Diagram for an overlapping numerically MC-minimal quadratic configuration.
• Figure 5: Diagram for the non-overlapping associated to Example \ref{['ex_overlapping']}.

Theorems & Definitions (50)

• Proposition 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Example 2.5
• Definition 2.6
• Theorem 2.7
• Theorem 2.8
• Lemma 2.9