Higgs bundles twisted by a vector bundle

Abstract

In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank 2 vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin-Kobayashi correspondence for a generalization of the Hitchin equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher dimensional variety.

23 pages. To appear in the International Journal of Mathematics, we have included some referee comments. We have added and expanded some remarks and comments. We have also added an extra case in Section 3.4.3