Asian Journal of Mathematics

Volume 21 (2017)

Number 5

Connectedness of Higgs bundle moduli for complex reductive Lie groups

Pages: 791 – 810



Oscar García-Prada (Instituto de Ciencias Matem´aticas, Madrid, Spain)

André Oliveira (Centro de Matemática da Universidade do Porto, Portugal; and Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, Vila Real, Portugal)


We carry an intrinsic approach to the study of the connectedness of the moduli space $\mathcal{M}_G$ of $G$-Higgs bundles, over a compact Riemann surface, when $G$ is a complex reductive (not necessarily connected) Lie group. We prove that the number of connected components of $\mathcal{M}_G$ is indexed by the corresponding topological invariants. In particular, this gives an alternative proof of the counting by J. Li in “The Space of Surface Group Representations” [Manuscripta Math., 78 (1993), pp. 223–243] of the number of connected components of the moduli space of flat $G$-connections in the case in which $G$ is connected and semisimple.


semistable Higgs bundles, connected components of moduli spaces

2010 Mathematics Subject Classification

14D20, 14F45, 14H60

Full Text (PDF format)

Received 20 August 2014

Accepted 17 March 2016

Published 9 February 2018