Co-Higgs bundles on P^1

Abstract

Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P^1, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank-2, odd-degree moduli space as a universal elliptic curve with a globally-defined equation. For ranks r=2,3,4, we explicitly verify the conjectural Betti numbers emerging from the recent work of Chuang, Diaconescu, Pan, and Mozgovoy on the ADHM formula. We state the result for r=5.

18 pages, 1 table. This version of the paper matches the published version. This version features expanded remarks, corrections, and two new sections on Betti numbers of moduli spaces that do not appear in the original preprint. (References to the original preprint should be unaffected, as all of its results are preserved in the new version.)

References in corpus (4)

• Mirror symmetry, Langlands duality, and the Hitchin system
• Dimensional reduction, SL(2,C)-equivariant bundles and stable holomorphic chains
• Generalized holomorphic bundles and the B-field action
• Geometry of the moduli space of Higgs bundles

Cited by in corpus (2)

• Higgs bundles over P^1 and quiver representations
• A Kobayashi-Hitchin correspondence for $I_{\pm }$-holomorphic bundles