Rigid non-cohomologically rigid local systems

TL;DR

This paper constructs new examples of irreducible rigid local systems of various ranks on algebraic varieties, highlighting differences between cohomological rigidity and other forms of rigidity, with implications for monodromy representations.

Contribution

It introduces explicit constructions of irreducible rigid local systems that are not cohomologically rigid, expanding the understanding of rigidity in local systems on algebraic varieties.

Findings

Constructed irreducible rigid local systems of rank r for any even r ≥ 2.

Provided examples of local systems that are rigid but not cohomologically rigid.

Extended examples to include systems with infinite monodromy by exterior product.

Abstract

For any even natural number $r \ge 2$, we construct an irreducible rigid non-cohomologically rigid complex local system of rank $r$ on a smooth projective variety depending on $r$. For $r=2$, we construct an irreducible rigid non-cohomogically rigid local system of rank $2$ on a quasi-projective variety which becomes cohomologically rigid after fixing the conjugacy classes of the monodromies at infinity. v2: We added a remark due to Alexander Petrov: by taking the exterior product of our examples with a (cohomologically) rigid local system with infinite monodromy, we obtain examples of rigid non-cohomologically rigid local systems with infinite monodromy.