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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.06267 |
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| _version_ | 1866910563733340160 |
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| author | Hallam, Michael Lahdili, Abdellah |
| author_facet | Hallam, Michael Lahdili, Abdellah |
| contents | We introduce a new weighted version of the Hermite--Einstein equation, along with notions of weighted slope (semi/poly)stability, and prove that a vector bundle admits a weighted Hermite--Einstein metric if and only if it is weighted slope polystable. The new equation encompasses several well-known examples of canonical Hermitian metrics on vector bundles, including the usual Hermite--Einstein metrics, Kähler--Ricci solitons, and transversally Hermite--Einstein metrics on certain Sasaki manifolds. We prove that the equation arises naturally as a moment map, that solutions to the equation are unique up to scaling, and demonstrate a weighted Kobayashi--Lübke inequality satisfied by vector bundles admitting a weighted Hermite--Einstein metric. As an application of our techniques, we extend a bound of Tian on the Ricci curvature to a bound on a modified Ricci curvature, related to the existence of Kähler--Ricci solitons. Along the way, we introduce a new weighted vortex equation, as well as a weighted analogue of Gieseker stability. A key technical point is the application of a new extension of Inoue's equivariant intersection numbers to arbitrary weight functions on the moment polytope of a Kähler manifold with Hamiltonian torus action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06267 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The weighted Hermite--Einstein equation Hallam, Michael Lahdili, Abdellah Differential Geometry Algebraic Geometry 53C07, 32L05 We introduce a new weighted version of the Hermite--Einstein equation, along with notions of weighted slope (semi/poly)stability, and prove that a vector bundle admits a weighted Hermite--Einstein metric if and only if it is weighted slope polystable. The new equation encompasses several well-known examples of canonical Hermitian metrics on vector bundles, including the usual Hermite--Einstein metrics, Kähler--Ricci solitons, and transversally Hermite--Einstein metrics on certain Sasaki manifolds. We prove that the equation arises naturally as a moment map, that solutions to the equation are unique up to scaling, and demonstrate a weighted Kobayashi--Lübke inequality satisfied by vector bundles admitting a weighted Hermite--Einstein metric. As an application of our techniques, we extend a bound of Tian on the Ricci curvature to a bound on a modified Ricci curvature, related to the existence of Kähler--Ricci solitons. Along the way, we introduce a new weighted vortex equation, as well as a weighted analogue of Gieseker stability. A key technical point is the application of a new extension of Inoue's equivariant intersection numbers to arbitrary weight functions on the moment polytope of a Kähler manifold with Hamiltonian torus action. |
| title | The weighted Hermite--Einstein equation |
| topic | Differential Geometry Algebraic Geometry 53C07, 32L05 |
| url | https://arxiv.org/abs/2408.06267 |