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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2401.07759 |
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| _version_ | 1866916387313680384 |
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| author | Silva, Pedro M. Gothen, Peter B. |
| author_facet | Silva, Pedro M. Gothen, Peter B. |
| contents | The non-abelian Hodge correspondence maps a polystable $\mathrm{SL}(2,\mathbb{R})$-Higgs bundle on a compact Riemann surface $X$ of genus $g\geq2$ to a connection which, in some cases, is the holonomy of a branched hyperbolic structure. On the other hand, Gaiotto's conformal limit maps the same bundle to a partial oper, i.e., to a connection whose holonomy is that of a branched complex projective structure compatible with $X$.
In this article, we show how these are both instances of the same phenomenon: the family of connections appearing in the conformal limit can be understood as a family of complex projective structures, deforming the hyperbolic ones into the ones compatible with $X$.
We also show that, when the Higgs bundle has zero Toledo invariant, this deformation is optimal, inducing a geodesic on Teichmüller's metric space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07759 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The conformal limit and projective structures Silva, Pedro M. Gothen, Peter B. Differential Geometry Algebraic Geometry 30F10 (Primary) 14D21, 14H60, 53C43 (Secondary) The non-abelian Hodge correspondence maps a polystable $\mathrm{SL}(2,\mathbb{R})$-Higgs bundle on a compact Riemann surface $X$ of genus $g\geq2$ to a connection which, in some cases, is the holonomy of a branched hyperbolic structure. On the other hand, Gaiotto's conformal limit maps the same bundle to a partial oper, i.e., to a connection whose holonomy is that of a branched complex projective structure compatible with $X$. In this article, we show how these are both instances of the same phenomenon: the family of connections appearing in the conformal limit can be understood as a family of complex projective structures, deforming the hyperbolic ones into the ones compatible with $X$. We also show that, when the Higgs bundle has zero Toledo invariant, this deformation is optimal, inducing a geodesic on Teichmüller's metric space. |
| title | The conformal limit and projective structures |
| topic | Differential Geometry Algebraic Geometry 30F10 (Primary) 14D21, 14H60, 53C43 (Secondary) |
| url | https://arxiv.org/abs/2401.07759 |