Canonical metrics on families of vector bundles
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914178933981184 |
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| author | Lam, Shing Tak |
| author_facet | Lam, Shing Tak |
| contents | We introduce a geometric partial differential equation for families of holomorphic vector bundles, generalising the theory of Hermite--Einstein metrics. We consider families of holomorphic vector bundles which each admit Hermite--Einstein metrics, together with a first order deformation. On such families, we define the family Hermite--Einstein equation for Hermitian metrics, which we view as a notion of a canonical metric in this setting.
We prove two main results concerning family Hermite--Einstein metrics. Firstly, we construct Hermite--Einstein metrics in adiabatic classes on product manifolds, assuming the existence of a family Hermite--Einstein metric. Secondly, we prove that the associated parabolic flow admits a unique smooth solution for all time, and use this to show that the Dirichlet problem always admits a unique solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canonical metrics on families of vector bundles Lam, Shing Tak Differential Geometry Algebraic Geometry We introduce a geometric partial differential equation for families of holomorphic vector bundles, generalising the theory of Hermite--Einstein metrics. We consider families of holomorphic vector bundles which each admit Hermite--Einstein metrics, together with a first order deformation. On such families, we define the family Hermite--Einstein equation for Hermitian metrics, which we view as a notion of a canonical metric in this setting. We prove two main results concerning family Hermite--Einstein metrics. Firstly, we construct Hermite--Einstein metrics in adiabatic classes on product manifolds, assuming the existence of a family Hermite--Einstein metric. Secondly, we prove that the associated parabolic flow admits a unique smooth solution for all time, and use this to show that the Dirichlet problem always admits a unique solution. |
| title | Canonical metrics on families of vector bundles |
| topic | Differential Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2512.04017 |