Taiwanese Journal of Mathematics

A Note on co-Higgs Bundles

Edoardo Ballico and Sukmoon Huh

Full-text: Open access

Abstract

We show that for any ample line bundle on a smooth complex projective variety with nonnegative Kodaira dimension, the semistability of co-Higgs bundles of implies the semistability of bundles. Then we investigate the criterion for surface $X$ to have $H^0(T_X) = H^0(S^2 T_X) = 0$, which implies that any co-Higgs structure of rank two is nilpotent.

Article information

Source
Taiwanese J. Math., Volume 21, Number 2 (2017), 267-281.

Dates
First available in Project Euclid: 29 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498750952

Digital Object Identifier
doi:10.11650/tjm/7834

Mathematical Reviews number (MathSciNet)
MR3632515

Zentralblatt MATH identifier
06871317

Subjects
Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
Secondary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 53D18: Generalized geometries (à la Hitchin)

Keywords
co-Higgs bundle stability global tangent vector field

Citation

Ballico, Edoardo; Huh, Sukmoon. A Note on co-Higgs Bundles. Taiwanese J. Math. 21 (2017), no. 2, 267--281. doi:10.11650/tjm/7834. https://projecteuclid.org/euclid.twjm/1498750952


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References

  • W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact Complex Surfaces, Second edition, Springer-Verlag, Berlin, 2004.
  • A. V. Colmenares, Semistable Rank $2$ co-Higgs Bundles over Hirzebruch Surfaces, Ph.D. thesis, Waterloo, Ontario, 2015.
  • ––––, Moduli spaces of semistable rank-$2$ co-Higgs bundles over $\mb{P}^1 \times \mb{P}^1$. arXiv:1604.01372
  • M. Corrêa, Rank two nilpotent co-Higgs sheaves on complex surfaces, Geom. Dedicata 183 (2016), no. 1, 25–31.
  • W. Fulton, Intersection Theory, Springer-Verlag, Berlin, 1984.
  • M. Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75–123.
  • N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308.
  • D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics E31, Friedr. Vieweg & Sohn, Braunschweig, 1997.
  • R. Lazarsfeld, Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and Multiplier Ideals, Springer-Verlag, Berlin, 2004.
  • V. B. Mehta and A. Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann. 258 (1982), no. 3, 213–224.
  • Y. Miyaoka, Algebraic surfaces with positive indices, in Classification of Algebraic and Analytic Manifolds, (Katata, 1982), 281–301, Progr. Math. 39, Birkhäuser Boston, Boston, MA, 1983.
  • T. Peternell, Generically nef vector bundles and geometric applications, in Complex and Differential Geometry, 345–368, Springer Proc. Math. 8, Springer, Heidelberg, 2011.
  • S. Rayan, Geometry of co-Higgs Bundles, Ph.D. thesis, Oxford, 2011.
  • ––––, Constructing co-Higgs bundles on $\mb{CP}^2$, Q. J. Math. 65 (2014), no. 4, 1437–1460.
  • N. I. Shepherd-Barron, Miyaoka's theorems on the generic seminegativity of $T_X$ and on the Kodaira dimension of minimal regular threefolds, in Flips and Abundance for Algebraic Threefolds, Asterisque, 211 (1992), 103–114.