# System of Hodge Bundles and Generalized Opers on Smooth Projective Varieties

## TL;DR

This paper establishes criteria for the stability of Hodge bundles and introduces generalized opers on smooth projective varieties, proving their associated Higgs bundles are semistable and linking oper structures to bundle stability.

## Contribution

It defines generalized opers on smooth projective varieties and proves their associated Higgs bundles are semistable, also connecting partial oper structures with bundle stability.

## Key findings

- Semistability criteria for Hodge bundles established
- Generalized opers' associated Higgs bundles are semistable
- Partial oper structures imply semistability of (E, ∇) pairs

## Abstract

Let $k$ be an algebraically closed field of any characteristic. Let $X$ be a polarized irreducible smooth projective algebraic variety over $k$. We give criterion for semistability and stability of system of Hodge bundles on $X$. We define notion of generalized opers on $X$, and prove semistability of the Higgs bundle associated to generalized opers. We also show that existence of partial oper structure on a vector bundle $E$ together with a connection $\nabla$ over $X$ implies semistability of the pair $(E, \nabla)$.