Semistable Higgs bundles on elliptic surfaces
-
Ugo Bruzzo
and
Vitantonio Peragine
Abstract
We analyze Higgs bundles (V, ϕ) on a class of elliptic surfaces π : 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 ϕ 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 π (respectively, if the fundamental line bundle is sufficiently ample), then ϕ 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.
Acknowledgements
The second author wishes to thank the Department of Mathematics of Universidade Federal da Paraíba, João Pessoa, for hospitality; and Valeriano Lanza for the invitation to give a talk on a preliminary version of {this} paper at Universidade Federal Fluminense, Rio de Janeiro.
Communicated by: G. Gentili
Funding: U.B.’s research is partly supported by PRIN “Geometria delle varietà algebriche” and INdAM-GNSAGA. He is a member of the VBAC group.
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Articles in the same Issue
- Frontmatter
- Semistable Higgs bundles on elliptic surfaces
- On the rigidity of harmonic-Ricci solitons
- On Ricci negative derivations
- Approximating coarse Ricci curvature on submanifolds of Euclidean space
- Convex cones spanned by regular polytopes
- On the geography of line arrangements
- Towards resolving Keller’s cube tiling conjecture in dimension seven
Articles in the same Issue
- Frontmatter
- Semistable Higgs bundles on elliptic surfaces
- On the rigidity of harmonic-Ricci solitons
- On Ricci negative derivations
- Approximating coarse Ricci curvature on submanifolds of Euclidean space
- Convex cones spanned by regular polytopes
- On the geography of line arrangements
- Towards resolving Keller’s cube tiling conjecture in dimension seven
