Continuity of HYM connections with respect to metric variations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912447965691904 |
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| author | Delloque, Rémi |
| author_facet | Delloque, Rémi |
| contents | We investigate the set of (real Dolbeault classes of) balanced metrics $Θ$ on a balanced manifold $X$ with respect to which a torsion-free coherent sheaf $\mathcal{E}$ on $X$ is slope stable. We prove that the set of all such $[Θ] \in H^{n - 1,n - 1}(X,\mathbb{R})$ is an open convex cone locally defined by a finite number of linear inequalities.
When $\mathcal{E}$ is a Hermitian vector bundle, the Kobayashi--Hitchin correspondence provides associated Hermitian Yang--Mills connections, which we show depend continuously on the metric, even around classes with respect to which $\mathcal{E}$ is only semi-stable. In this case, the holomorphic structure induced by the connection is the holomorphic structure of the associated graded object. The method relies on semi-stable perturbation techniques for geometric PDEs with a moment map interpretation and is quite versatile, and we hope that it can be used in other similar problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuity of HYM connections with respect to metric variations Delloque, Rémi Differential Geometry Algebraic Geometry 53C07 (primary), 35B20 (secondary) We investigate the set of (real Dolbeault classes of) balanced metrics $Θ$ on a balanced manifold $X$ with respect to which a torsion-free coherent sheaf $\mathcal{E}$ on $X$ is slope stable. We prove that the set of all such $[Θ] \in H^{n - 1,n - 1}(X,\mathbb{R})$ is an open convex cone locally defined by a finite number of linear inequalities. When $\mathcal{E}$ is a Hermitian vector bundle, the Kobayashi--Hitchin correspondence provides associated Hermitian Yang--Mills connections, which we show depend continuously on the metric, even around classes with respect to which $\mathcal{E}$ is only semi-stable. In this case, the holomorphic structure induced by the connection is the holomorphic structure of the associated graded object. The method relies on semi-stable perturbation techniques for geometric PDEs with a moment map interpretation and is quite versatile, and we hope that it can be used in other similar problems. |
| title | Continuity of HYM connections with respect to metric variations |
| topic | Differential Geometry Algebraic Geometry 53C07 (primary), 35B20 (secondary) |
| url | https://arxiv.org/abs/2403.16814 |