Flow lines on the moduli space of rank $2$ twisted Higgs bundles
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912365176422400 |
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| author | Wilkin, Graeme |
| author_facet | Wilkin, Graeme |
| contents | This paper studies the gradient flow lines for the $L^2$ norm square of the Higgs field defined on the moduli space of semistable rank $2$ Higgs bundles twisted by a line bundle of positive degree over a compact Riemann surface $X$. The main result is that these spaces of flow lines have an algebro-geometric classification in terms of secant varieties for different embeddings of $X$ into the projectivisation of the negative eigenspace of the Hessian at a critical point. The Morse-theoretic compactification of spaces of flow lines given by adding broken flow lines then has a natural algebraic interpretation via a projection to Bertram's resolution of secant varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13098 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flow lines on the moduli space of rank $2$ twisted Higgs bundles Wilkin, Graeme Algebraic Geometry Differential Geometry 58D15 This paper studies the gradient flow lines for the $L^2$ norm square of the Higgs field defined on the moduli space of semistable rank $2$ Higgs bundles twisted by a line bundle of positive degree over a compact Riemann surface $X$. The main result is that these spaces of flow lines have an algebro-geometric classification in terms of secant varieties for different embeddings of $X$ into the projectivisation of the negative eigenspace of the Hessian at a critical point. The Morse-theoretic compactification of spaces of flow lines given by adding broken flow lines then has a natural algebraic interpretation via a projection to Bertram's resolution of secant varieties. |
| title | Flow lines on the moduli space of rank $2$ twisted Higgs bundles |
| topic | Algebraic Geometry Differential Geometry 58D15 |
| url | https://arxiv.org/abs/2408.13098 |