The behavior of sequences of solutions to the Hitchin-Simpson equations
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arXiv
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| Format: | Preprint |
| Published: |
2020
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| _version_ | 1866916452075831296 |
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| author | He, Siqi |
| author_facet | He, Siqi |
| contents | The Hitchin-Simpson equations are first-order non-linear equations for a pair consisting of a connection and a Higgs field. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations on closed Kähler manifolds with unbounded $L^2$ norms of the Higgs fields. We prove a compactness result for the connections and renormalized Higgs fields, which generalizes the work of Taubes and Mochizuki.
As applications, we prove that every $\mathbb{Z}/2$ harmonic 1-form on a Kähler manifold can be deformed into a sequence of solutions to the Hitchin-Simpson equations. Additionally, we solve the generalized Hitchin's WKB problem on any Kähler manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_08109 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The behavior of sequences of solutions to the Hitchin-Simpson equations He, Siqi Differential Geometry 53C07 The Hitchin-Simpson equations are first-order non-linear equations for a pair consisting of a connection and a Higgs field. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations on closed Kähler manifolds with unbounded $L^2$ norms of the Higgs fields. We prove a compactness result for the connections and renormalized Higgs fields, which generalizes the work of Taubes and Mochizuki. As applications, we prove that every $\mathbb{Z}/2$ harmonic 1-form on a Kähler manifold can be deformed into a sequence of solutions to the Hitchin-Simpson equations. Additionally, we solve the generalized Hitchin's WKB problem on any Kähler manifold. |
| title | The behavior of sequences of solutions to the Hitchin-Simpson equations |
| topic | Differential Geometry 53C07 |
| url | https://arxiv.org/abs/2002.08109 |