Semistable Higgs bundles on elliptic surfaces

Abstract We analyze Higgs bundles ( V , ϕ ) on a class of elliptic surfaces π : X → B whose underlying vector bundle has vertical determinant and is fiberwise semistable. prove that if the spectral curve reduced, then field while regular with reduced (respectively, integral) curve, its rank second Chern number satisfy an inequality involving genus degree fundamental line sufficiently ample), scalar . apply these results to problem characterizing slope-semistable vanishing discriminant considered, in terms semistability their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves