System of Hodge Bundles and Generalized Opers on Smooth Projective Varieties

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 $\mathrm{\nabla }$ over $X$ implies semistability of the pair $(E,\mathrm{\nabla })$.