Rank 3 rigid representations of projective fundamental groups

TL;DR

This paper proves that all rigid integral irreducible representations of rank 3 of the fundamental group of a smooth complex projective variety originate from geometric families, extending previous results from rank 2.

Contribution

It establishes that rank 3 rigid integral irreducible representations are of geometric origin, generalizing earlier rank 2 results and addressing a key open question.

Findings

All such representations are of geometric origin.

Extension of previous rank 2 results to rank 3.

Answers a question posed by Corlette and the second author.

Abstract

Let X be a smooth complex projective variety with basepoint x. We prove that every rigid integral irreducible representation $\pi_1(X,x)\to SL (3,{\mathbb C})$ is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by K. Corlette and the second author in the rank 2 case and answers one of their questions.