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 del over X implies semistability of the pair (E, del). (C) 2019 Elsevier B.V. All rights reserved.