CONSTRUCTING CO-HIGGS BUNDLES ON ℂℙ2

On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's rank-2 vector bundle on the projective plane, constructed from a line bundle on the double cover ℂℙ¹×ℂℙ¹→ℂℙ², is naturally a co-Higgs bundle, with the twisted endomorphism, or ‘Higgs field’, also descending from the double cover. Allowing the branch conic to vary, we find that Schwarzenberger bundles give rise to an eight-dimensional moduli space of co-Higgs bundles. After studying the deformation theory for co-Higgs bundles on complex manifolds, we conclude that a co-Higgs bundle arising from a Schwarzenberger bundle with non-zero Higgs field is rigid, in the sense that a nearby deformation is again Schwarzenberger.