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arXiv · 2408.06267

The weighted Hermite--Einstein equation

Michael Hallam·Abdellah Lahdili
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Abstract

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.

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Michael Hallam, Abdellah Lahdili. 2024-08-12. The weighted Hermite--Einstein equation. https://arxiv.org/abs/2408.06267

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