Absolute sets of rigid local systems

TL;DR

This paper investigates the conjecture that absolute sets of simple cohomologically rigid local systems are of geometric origin, reducing the problem to the zero-dimensional case and confirming it for curves and rank two cases.

Contribution

It reduces the conjecture for absolute sets to Simpson's conjecture for zero-dimensional cases and verifies it for curves and rank two local systems.

Findings

Conjecture reduces to Simpson's conjecture in zero-dimensional case

Confirmed conjecture for curves and rank two local systems

Provides a new approach to understanding absolute sets of local systems

Abstract

The absolute sets of local systems on a smooth complex algebraic variety are the subject of a conjecture of N. Budur and B. Wang based on an analogy with special subvarieties of Shimura varieties. An absolute set should be the higher-dimensional generalization of a local system of geometric origin. We show that the conjecture for absolute sets of simple cohomologically rigid local systems reduces to the zero-dimensional case, that is, to Simpson's conjecture that every such local system with quasi-unipotent monodromy at infinity and determinant is of geometric origin. In particular, the conjecture holds for this type of absolute sets if the variety is a curve or if the rank is two.