Symmetric powers and a problem of Kollar and Larsen

TL;DR

This paper proves a conjecture regarding the structure of certain subgroups of general linear groups acting irreducibly on symmetric powers, with implications for holonomy groups of vector bundles.

Contribution

It establishes the conjecture of Kollar and Larsen for symmetric powers with k ≥ 4, advancing understanding of subgroup actions on vector spaces.

Findings

Proves Kollar and Larsen's conjecture for symmetric powers with k ≥ 4.

Implications for the holonomy groups of stable vector bundles.

Connects subgroup actions to geometric properties of vector bundles.

Abstract

We prove a conjecture of Kollar and Larsen on Zariski closed subgroups of $GL(V)$ which act irreducibly on some symmetric power $Sym^{k}(V)$ with $k \geq 4$. This conjecture has interesting implications, in particular on the holonomy group of a stable vector bundle on a smooth projective variety, as shown by the recent work of Balaji and Kollar.