Semistable Higgs bundles on elliptic surfaces

We analyze Higgs bundles on a class of elliptic surfaces, whose underlying vector bundle has vertical determinant and is fiberwise semistable. We prove that if the spectral curve of the bundle is reduced, then the integrability condition for the Higgs field is automatically satisfied, and the Higgs field takes values in the pull-back of the canonical line bundle of the base curve of the elliptic fibration. We then prove that if the bundle is fiberwise regular with reduced (respectively, integral) spectral curve, and if its rank and second Chern number satisfy an inequality involving the genus of the base and the degree of the fundamental line bundle of the fibration (respectively, if the fundamental line bundle is sufficiently ample), then the Higgs fields acts by tensoring with a global 1-form on the surface. We apply these results to the problem of characterizing slope-semistable Higgs bundles with vanishing discriminant on the surface, in terms of the semistability of their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves to the surface.