rigid local systems

The goal of this seminar is to learn about Katz’s classification of rigid local systems on \(\mathbb{P}^1\setminus\{x_1, \cdots, x_n\}\), loosely following Volklein’s formalism in his paper The Braid Group and Linear Rigidity. The seminar met most Fridays at 11am in BA6180.

• Week 1 (January 27, 2023): Introduction (Daniel)

  • Introduction to the problem, the nature of Katz’s classification, some history, and the middle convolution.

• Week 2 (February 10, 2023): The braid group, following Chapter 1 of Birman’s “Braids, Links, and Mapping Class Groups.” (Kai)

  • The braid group via pictures, generators and relations, and the fundamental group of configuration space. The Birman exact sequence for the braid group and the induced outer action of the braid group on the free group. The action of the braid group on solutions to the Deligne-Simpson problem (see Volklein). Rigid local systems have finite orbit under the braid group action.

• Week 3 (February 17, 2023): Character varieties, following Sikora’s survey, taking \(G=GL_n\). (Charlie)

  • Representation and character varieties with examples. The tangent space to an irreducible representation in the character variety via cohomology of the adjoint representation. Dimension of the tangent space when the group is a surface group. Dimension of the tangent space when one fixes local monodromy at infinity.

• Week 4 (February 24, 2023): First properties of rigid local systems in genus zero, following Chapter 1 of Katz’s book. (Sasha)

  • Criterion for rigidity in terms of Euler characteristic. Rigid local systems are uniquely determined by their local monodromy. They exist only in genus zero. Time permitting, some discussion of the genus one situation.

• Week 5 (March 3, 2023): Braid companions, following section 1 (mostly 1.2) and 2.1 of Volklein’s paper (Jeremy)

  • The braid companion functor and basic properties. Equivariance for the braid group action. Explicit description of the braid companion. An example for two-dimensional representations.

• Week 6 (March 10, 2023): Katz’s classification, following section 2 of Volklein’s paper (Andy)

  • Completing the proof, focusing on 2.11 and 2.12.

• Further topics: applications to the inverse Galois problem, comparison of Volklein’s BC-companion to Katz’s middle convolution, Belkale’s work on unitary rigid local systems.