Semistable Higgs bundles on elliptic surfaces

We analyze Higgs bundles (V, phi) on a class of elliptic surfaces pi : X -> B, whose underlying vector bundle V has vertical determinant and is fiberwise semistable. We prove that if the spectral curve of V is reduced, then the Higgs field phi is vertical, while if the bundle V is fiberwise regular with reduced (respectively, integral) spectral curve, and if its rank and second Chern number satisfy an inequality involving the genus of B and the degree of the fundamental line bundle of pi (respectively, if the fundamental line bundle is sufficiently ample), then phi is scalar. 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 to X.