$Z$-critical connections and Bridgeland stability conditions
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866916098307260416 |
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| author | Dervan, Ruadhaí McCarthy, John Benjamin Sektnan, Lars Martin |
| author_facet | Dervan, Ruadhaí McCarthy, John Benjamin Sektnan, Lars Martin |
| contents | We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations $Z$-critical connections, with $Z$ a central charge. Deformed Hermitian Yang--Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps.
Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a $Z$-critical connection if and only if it is asymptotically $Z$-stable. Even for the deformed Hermitian Yang--Mills equation, this provides the first examples of solutions in higher rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_10426 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | $Z$-critical connections and Bridgeland stability conditions Dervan, Ruadhaí McCarthy, John Benjamin Sektnan, Lars Martin Differential Geometry Algebraic Geometry We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations $Z$-critical connections, with $Z$ a central charge. Deformed Hermitian Yang--Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps. Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a $Z$-critical connection if and only if it is asymptotically $Z$-stable. Even for the deformed Hermitian Yang--Mills equation, this provides the first examples of solutions in higher rank. |
| title | $Z$-critical connections and Bridgeland stability conditions |
| topic | Differential Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2012.10426 |