The Algebraic and Analytic Compactifications of the Hitchin Moduli Space
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910725474091008 |
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| author | He, Siqi Mazzeo, Rafe Na, Xuesen Wentworth, Richard |
| author_facet | He, Siqi Mazzeo, Rafe Na, Xuesen Wentworth, Richard |
| contents | Following the work of Mazzeo-Swoboda-Weiss-Witt and Mochizuki, there is a map $\overlineΞ$ between the algebraic compactification of the Dolbeault moduli space of $\mathsf{SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the $\mathbb{C}^\ast$ action, and the analytic compactification of Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ``limiting configurations''. This map extends the classical Kobayashi-Hitchin correspondence. The main result of this paper is that $\overlineΞ$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. This suggests the possibility of a third, refined compactification which dominates both. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_08198 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Algebraic and Analytic Compactifications of the Hitchin Moduli Space He, Siqi Mazzeo, Rafe Na, Xuesen Wentworth, Richard Differential Geometry 32G13, 53C07 (primary), 14D20 (secondary) Following the work of Mazzeo-Swoboda-Weiss-Witt and Mochizuki, there is a map $\overlineΞ$ between the algebraic compactification of the Dolbeault moduli space of $\mathsf{SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the $\mathbb{C}^\ast$ action, and the analytic compactification of Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ``limiting configurations''. This map extends the classical Kobayashi-Hitchin correspondence. The main result of this paper is that $\overlineΞ$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. This suggests the possibility of a third, refined compactification which dominates both. |
| title | The Algebraic and Analytic Compactifications of the Hitchin Moduli Space |
| topic | Differential Geometry 32G13, 53C07 (primary), 14D20 (secondary) |
| url | https://arxiv.org/abs/2304.08198 |