[Paper Review] On exceptional rigid local systems

Michael Dettweiler, Stefan Reiter|ArXiv.org|Sep 5, 2006

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper proves new instances of Simpson’s rigidity conjecture by constructing a $G_2$-rigid local system on the four-punctured Riemann sphere that is motivic despite not being $\mathrm{GL}_7$-rigid. Using tensor products, middle convolution, and the Künneth formula, the authors show the local system arises from a higher direct image of a smooth morphism, confirming its motivic nature and extending known results beyond $\mathrm{GL}_n$-rigid cases.

ABSTRACT

We prove new instances of Simpson's rigidity conjecture which states that quasi-unipotent rigid local systems should be motivic. We construct new relative motives over the fourfold punctured Riemann sphere which give rise to $G_2$-rigid local systems which are not rigid in the group $\GL_7.$

Motivation & Objective

• To verify Simpson’s rigidity conjecture for $G_2$-rigid local systems that are not $\mathrm{GL}_7$-rigid.
• To extend the known class of motivic local systems beyond those arising from $\mathrm{GL}_n$-rigidity.
• To construct explicit $G_2$-rigid local systems on $\mathbb{P}^1 \setminus \{x_1,x_2,x_3,\infty\}$ with prescribed monodromy Jordan forms.
• To demonstrate that $G_2$-rigidity does not imply $\mathrm{GL}_7$-rigidity, yet the rigidity conjecture still holds.

Proposed method

• Start with two $\mathrm{GL}_2(\mathbb{C})$-rigid local systems and take their tensor product to obtain a $\mathrm{SO}_4(\mathbb{C})$-rigid local system of rank 4.
• Apply middle convolution with parameter $\zeta_3^{-1}$ to the tensor product to obtain a new local system of rank 7.
• Use a second middle convolution and tensor operations to construct a $G_2(\mathbb{C})$-rigid local system ${\cal H}$ on $\mathbb{P}^1 \setminus \{x_1,x_2,x_3,\infty\}$.
• Verify that the monodromy tuple of ${\cal H}$ has the specified Jordan canonical forms: $\mathrm{J}(2,2,1,1,1)$, $\mathrm{J}(2,2,1,1,1)$, $\mathrm{diag}(1,\zeta_3,\zeta_3,\zeta_3,\zeta_3^{-1},\zeta_3^{-1},\zeta_3^{-1})$, and $\mathrm{J}(3,3,1)$.
• Confirm that the Zariski closure of the monodromy image is $G_2(\mathbb{C})$ using the structure of the monodromy matrices and group-theoretic constraints.
• Establish motivicity via the Künneth formula and the motivic interpretation of middle convolution from Katz’s work.

Experimental results

Research questions

• RQ1 Can $G_2$-rigid local systems that are not $\mathrm{GL}_7$-rigid still be motivic, as predicted by Simpson’s rigidity conjecture?
• RQ2 What is the monodromy structure of such $G_2$-rigid local systems on the four-punctured sphere?
• RQ3 Can the middle convolution and tensor operations be used to construct new motivic local systems beyond the $\mathrm{GL}_n$-rigid framework?
• RQ4 Is there a motivic interpretation for $G_2$-rigid local systems with non-trivial Jordan forms and non-irreducible $\mathrm{GL}_7$-conjugacy?
• RQ5 How do the weight and Hodge structure of such local systems compare to previously known examples?

Key findings

• A $G_2(\mathbb{C})$-rigid local system ${\cal H}$ exists on $\mathbb{P}^1 \setminus \{x_1,x_2,x_3,\infty\}$ with monodromy Jordan forms $\mathrm{J}(2,2,1,1,1)$, $\mathrm{J}(2,2,1,1,1)$, $\mathrm{diag}(1,\zeta_3,\zeta_3,\zeta_3,\zeta_3^{-1},\zeta_3^{-1},\zeta_3^{-1})$, and $\mathrm{J}(3,3,1)$ at the four punctures.
• The Zariski closure of the monodromy representation of ${\cal H}$ is precisely $G_2(\mathbb{C})$, confirming its $G_2$-rigidity.
• The local system ${\cal H}$ is $\mathrm{SO}_7(\mathbb{C})$-rigid but not $\mathrm{GL}_7(\mathbb{C})$-rigid, showing that $G_2$-rigidity does not imply $\mathrm{GL}_7$-rigidity.
• The rigidity conjecture holds for ${\cal H}$, as it arises as a subfactor of a higher direct image $R^i\pi_*(\mathbb{C})$ via the motivic interpretation of middle convolution.
• The weight of ${\cal H}$ as a variation of Hodge structures is 4, contrasting with the weight 6 of the $G_2$-local system in earlier work.
• The explicit monodromy matrices $h_1, h_2, h_3 \in \mathrm{GL}_7(\mathbb{C})$ are provided, with $h_4 = (h_1 h_2 h_3)^{-1}$, and satisfy the required Jordan forms and group constraints.