Semistable Higgs bundles on elliptic surfaces

Abstract

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