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SIGMA, 2024 Volume 20, 111, 21 pp. (Mi sigma2113)

Harmonic Metrics for Higgs Bundles of Rank 3in the Hitchin Section

Hitoshi Fujioka

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Abstract: Given a tuple of holomorphic differentials on a Riemann surface, one can define a Higgs bundle in the Hitchin section and a natural symmetric pairing of the Higgs bundle. We study whether a Higgs bundle of rank 3 in the Hitchin section has a compatible harmonic metric when the spectral curve is a 2-sheeted branched covering of the Riemann surface. In particular, we give a condition for Higgs bundles in the Hitchin section on $\mathbb{C}$ or $\mathbb{C}^*$ to have compatible harmonic metrics.

Keywords: Higgs bundles, Hitchin section, harmonic metrics.

MSC: 53C07, 14D21

Received: April 21, 2024; in final form November 29, 2024; Published online December 11, 2024

Language: English

DOI: 10.3842/SIGMA.2024.111


ArXiv: 2403.07258


© Steklov Math. Inst. of RAS, 2025